Hilchot Tefilla - הַחֹ֧דֶשׁ הַזֶּ֛ה and חָמֵץ
According to Rashi, the Torah really should have started with Parashat Hachodesh:

(א) בראשית. …לֹֹֹֹֹא הָיָה צָרִיךְ לְהַתְחִיל אֶת הַתּוֹרָה אֶלָּא מֵהַחֹדֶשׁ הַזֶּה לָכֶם, שֶׁהִיא מִצְוָה רִאשׁוֹנָה שֶׁנִּצְטַוּוּ בָּהּ יִשׂרָאֵל

(1) בראשית IN THE BEGINNING — Rabbi Isaac said: The Torah which is the Law book of Israel should have commenced with the verse (Exodus 12:2) “This month shall be unto you the first of the months” which is the first commandment given to Israel

So what is the special significance of Parashat Hachodesh?

(ב) הַחֹ֧דֶשׁ הַזֶּ֛ה לָכֶ֖ם רֹ֣אשׁ חֳדָשִׁ֑ים רִאשׁ֥וֹן הוּא֙ לָכֶ֔ם לְחׇדְשֵׁ֖י הַשָּׁנָֽה׃

(2) This month shall mark for you the beginning of the months; it shall be the first of the months of the year for you.

החדש הזה לכם ראש חדשים. מכאן ואילך יהיו החדשים שלכם, לעשות בהם כרצונכם, אבל בימי השעבוד לא היו ימיכם שלכם, אבל היו לעבודת אחרים ורצונם...

החודש הזה לכם ראש חדשים, from now on these months will be yours, to do with as you like. [you have My authority to organise your own calendar. Ed.] This is by way of contrast to the years when you were enslaved when you had no control over your time or timetable at all. [Freedom, i.e. retirement from the “rat race,” means being able to formulate one’s own timetable. Ed.] While you were enslaved, your days, hours, minutes even, were always at the beck and call of your taskmasters.

(א) החדש הזה. ראש לחירות ואותו תעשו ראשון למניין חדשיכם כדי שתמנו משעת החירות ובזאת תזכר לכם שעת החירות ותזכרו את הטובה שעשיתי עמכם

אָמַר רַב אַחָא בַּר יַעֲקֹב: וְצָרִיךְ שֶׁיַּזְכִּיר יְצִיאַת מִצְרַיִם בְּקִידּוּשׁ הַיּוֹם. כְּתִיב הָכָא: ״לְמַעַן תִּזְכּוֹר אֶת יוֹם״. וּכְתִיב הָתָם: ״זָכוֹר אֶת יוֹם הַשַּׁבָּת לְקַדְּשׁוֹ״.

Rav Aḥa bar Ya’akov said: And one must mention the exodus from Egypt in the kiddush of Shabbat day, despite the fact that Shabbat is not directly connected to the Exodus. The proof is that here, with regard to Passover, it is written: “That you may remember the day when you came out of the land of Egypt all the days of your life” (Deuteronomy 16:3); and it is written there, with regard to Shabbat: “Remember the Shabbat day to sanctify it” (Exodus 20:8). By means of a verbal analogy of the word “day,” these verses teach that one must also recall the Exodus on Shabbat.

ראשון הוא לכם מכאן ואילך הוא ראשון—שמתוך שתימנו החדשים מזמן החירות תזכרו הטובה שעשיתי עמכם—אבל מאדם הראשון עד עכשיו תשרי ראשון וניסן שביעי, כדכתיב במלכים "בְּיֶ֥רַח הָאֵֽתָנִ֖ים בֶּחָ֑ג" ומתרגם יונתן "בְּיַרְחָא דְעַתִּיקַיָא דְקָרָן לֵיהּ יַרְחָא קַדְמָאָה וּכְעַן הוּא יַרְחָא שְׁבִיעָאָה".

ראשון הוא לכם, “it is a first for you;” from now on this will be a first month for you, the reason being that you will start counting by months as a reminder of when you received your freedom from slavery. It will forever remind you of the Good that I have done for you. Up until now, the month of Tishrey had been the first month and Nissan the seventh. This agrees with what is written in Kings I, 8,2: בירח האיתנים בחג, “in the month of the original ones “ (the patriarchs, or in the months whom the early generations of mankind referred to) as the seventh month.”Yonatan ben Uzziel translates the word איתנים as “the ancient generations of man called it the first month and now it is the seventh month.”

וַיִּקָּ֨הֲל֜וּ אֶל־הַמֶּ֤לֶךְ שְׁלֹמֹה֙ כָּל־אִ֣ישׁ יִשְׂרָאֵ֔ל בְּיֶ֥רַח הָאֵֽתָנִ֖ים בֶּחָ֑ג ה֖וּא הַחֹ֥דֶשׁ הַשְּׁבִיעִֽי׃

The entire body of Israel gathered before King Solomon at the Feast [of Booths], in the month of Ethanim—that is, the seventh month.

וְאִתְכְּנִישׁוּ לְוַת מַלְכָּא שְׁלֹמֹה כָּל אֱנַשׁ יִשְׂרָאֵל בְּיַרְחָא דְעַתִּיקַיָא דְקָרָן לֵיהּ יַרְחָא קַדְמָאָה חַגָא וּכְעַן הוּא יַרְחָא שְׁבִיעָאָה:

Then everyone in Israel was gathered to King Solomon in the month that the ancients used to call the first month, the Feast,216 but now it is the seventh month.

Why are we not allowed to eat Chametz on Pesach, and commanded to eat Matzot?

Approach #1: we eat Matzot, as no 'Chametz may be turned into smoke as offering to God'
(ח) וְאָכְל֥וּ אֶת־הַבָּשָׂ֖ר בַּלַּ֣יְלָה הַזֶּ֑ה צְלִי־אֵ֣שׁ וּמַצּ֔וֹת עַל־מְרֹרִ֖ים יֹאכְלֻֽהוּ׃
(8) They shall eat the flesh that same night; they shall eat it roasted over the fire, with unleavened bread and with bitter herbs.

על מרורים יאכלוהו כמו עם, וכמוהו "ויבאו האנשים על הנשים" (שמות ל״ה:כ״ב), וכן "ראשו על כרעיו ועל קרבו" (שמות י״ב:ט׳)... וילמד הכתוב שאין מצוה במרורים אלא עם אכילת הבשר, אבל המצות חזר וצוה (שמות י״ב:י״ח) "בערב תאכלו מצות", אפילו בפני עצמן

AND UNLEAVENED BREAD, ‘AL’ (WITH) BITTER HERBS THEY SHALL EAT IT. The purport of the verse is as follows: And they shall eat the flesh in that night, roast with fire, and with unleavened bread with bitter herbs, they shall eat it. [The word al here thus means “with,” and not “upon,” as it generally does.] Similarly: ‘al’ unleavened bread and bitter herbs means “with”; And they came both men ‘al’ women [means “with” the women]; its head ‘al’ its legs ‘v’al’ the inwards thereof [means its head “with” its legs “and with” the inwards thereof]. Scripture does not say im (with) [but instead uses al] in order to instruct that it is not obligatory to wrap them together, [i.e., insert the paschal meat and the bitter herbs between the unleavened bread], and eat them. Thus [the word umatzoth (and unleavened bread)] is missing the letter beth, [which would make it ubematzoth (and with unleavened bread)], similar to the expressions: they shall wash ‘mayim’ (water) [the word missing a beth, which would make it b’mayim, (with water)]; And Seled died ‘lo’ (no) children, [the word lo missing a beth, which would make it b’lo (without) children]. Thus Scripture teaches that there is no commandment for eating the bitter herbs alone, but only with the eating of the flesh of the Passover-offering. Concerning unleavened bread, however, Scripture repeated it by commanding, At even ye shall eat unleavened bread, even by itself [when there is no Passover-offering], as is the opinion of our Rabbis.
A more correct interpretation is that we say that the word, umatzoth (and unleavened bread), is connected with the earlier part of the verse: And they shall eat the flesh … and unleavened bread. The verse then continues to command that the flesh mentioned be eaten with bitter herbs. Thus He commanded the eating of the flesh with the eating of the unleavened bread, but He did not command the eating of bitter herbs, except by saying that they should eat the meat with bitter herbs, thus hinting that there is no [separate] commandment regarding the bitter herbs. It is only that the meat must be eaten with them, and when there is no Passover-offering there is no specific commandment [of the Torah] regarding the eating of the bitter herbs. It also teaches us that the bitter herbs do not invalidate the meat. Thus, if one ate the meat of the Passover-offering and did not eat bitter herbs, he has [nevertheless] fulfilled his duty of eating the Passover-offering, since the commandment concerning the eating of the Passover-offering is like the commandment of eating unleavened bread, each one an independent commandment in itself.

כׇּל־הַמִּנְחָ֗ה אֲשֶׁ֤ר תַּקְרִ֙יבוּ֙ לַה' לֹ֥א תֵעָשֶׂ֖ה חָמֵ֑ץ כִּ֤י כׇל־שְׂאֹר֙ וְכׇל־דְּבַ֔שׁ לֹֽא־תַקְטִ֧ירוּ מִמֶּ֛נּוּ אִשֶּׁ֖ה לַֽה'׃

No meal offering that you offer to ה' shall be made with leaven, for no leaven or honey may be turned into smoke as an offering by fire to ה'.

כל המנחה, הן המנחות האמורות בפרשה זו, והן המנחות המפורשות במקום אחר כלם אסורות להעשות חמץ, חוץ משתי הלחם שבאין חמץ, גם מקצת לחמי תודה, אבל אין מהן דבר למזבח, ולכן אמר אשר תקריבו לה' כלומר אותן שיש מהן דבר למזבח:

Honey and leaven work in the natural mode of generation, sacrifice works in the divine mode… they are [to be] kept apart.

Antique bakery did not depend on packets of processed yeast. Leaven was kept in the dough, and at each baking a portion was reserved to start off the next batch. The leaven in yesterday's dough would be kept somewhere cool so that it would be quiescent until needed. Then honey and warm water and some new flour would be added to activate it again. Gradually the warmth and the sweetness reawaken the old dough and the lump starts to swell… it exemplifies the multiplying, sprouting, bringing forth, and increasing that was blessed in Genesis.

The reason for forbidding honey is the same as for leaven. There is nothing bad about either. They combine to create fermentation, and by that action they are examples of teeming life.

[Likewise] Leviticus commands the people not to eat blood, not to eat an animal that has died an unconsecrated death, that is, an animal that has died of itself, or an animal torn by beasts, presumably with its blood still in it (Lev 17: 8-16; see also Deut 14:21). The dietary laws thus support the law against unconsecrated killing. Leviticus’ reverential attitude to life, animal and human, explains the animal corpse pollution rules. Thou shalt not stand upon [profit from] another's blood (Lev 19: 16).

Covenant and fertility are two contrasted principles. Covenant gives the paradigm of the laws at Sinai and the tabernacle... The protective laws that tell humans to avoid teeming creatures demonstrate God's compassion. The balance between the divine attributes, justice and mercy, gives a more intelligible reading than does the idea of God's horror of impurity -- Mary Douglas, Leviticus as Literature

Approach #2: we eat Matzot, as we were in a hurry to leave Egypt...

(לד) וַיִּשָּׂ֥א הָעָ֛ם אֶת־בְּצֵק֖וֹ טֶ֣רֶם יֶחְמָ֑ץ מִשְׁאֲרֹתָ֛ם צְרֻרֹ֥ת בְּשִׂמְלֹתָ֖ם עַל־שִׁכְמָֽם...(לט) וַיֹּאפ֨וּ אֶת־הַבָּצֵ֜ק אֲשֶׁ֨ר הוֹצִ֧יאוּ מִמִּצְרַ֛יִם עֻגֹ֥ת מַצּ֖וֹת כִּ֣י לֹ֣א חָמֵ֑ץ כִּֽי־גֹרְשׁ֣וּ מִמִּצְרַ֗יִם וְלֹ֤א יָֽכְלוּ֙ לְהִתְמַהְמֵ֔הַּ וְגַם־צֵדָ֖ה לֹא־עָשׂ֥וּ לָהֶֽם׃

(34) So the people took their dough before it was leavened, their kneading bowls wrapped in their cloaks upon their shoulders... (39) And they baked unleavened cakes of the dough that they had taken out of Egypt, for it was not leavened, since they had been driven out of Egypt and could not delay; nor had they prepared any provisions for themselves.

(א) משארותם. כלי עץ. כמו טנאך ומשארתך

(1) THEIR KNEADING-TROUGHS. Misheret (kneading-trough) is a wooden vessel. Compare, mishartekha (thy kneading-trough) in thy basket and thy kneading-trough (Deut. 28:5)

הפרטים שבפס' ל"ד על הבצק ועל המשארות הצרורות בשמלותם של בני ישראל והנתונות על שכמם נראים לכאורה מוזרים ומיותרים. אבל נוכל להבינם אם נניח, שכוונת הכתוב להכין מעכשיו מה שייאמר להלן, בפס' ל"ט, על הבצק שלא החמיץ. בהוראות על פסח מצרים מדובר על אכילת מצות רק בלילה, ולא ביום שלאחריו. הואיל ובלילה אכלו בני ישראל רק מצות. לא היה להם שאור כשיצאו, וישא העם את בצקו טרם יחמץ, אבל רוצים היו לאכול לחם טרי בהגיעם לתחנה הראשונה. ולפיכך לקחו אתם את משארותם צרורות בשמלותם, כדי שימהר הבצק להחמיץ מפני היותו צרור בתוך השמלות, ושמו אותן על שכמם, כדי שגם חום גופם יועיל לשם כך. אחר כך ייאמר בפס' ל"ט שכל זה לא הספיק, וגם ביום המחרת לא אכלו בני ישראל אלא עוגות מצות. וזה מסופר כדי לקשור את עניין פסח דורות, הממשיך את אכילת המצות גם אחר הלילה הראשון, במה שקרה ביציאת מצרים.

Some interesting Minhagim related to 'מִשְׁאֲרֹתָ֛ם צְרֻרֹ֥ת בְּשִׂמְלֹתָ֖ם'

צררת בשמלתם לכך נהגו העולם לתת האפיקומן תחת המפה זכר לצרורות בשמלותם.

צרורות בשמלותם, “bundled up inside their garments.” This line is the reason why we have the practice of wrapping the afikomen” in a serviette or tablecloth after breaking it and saving it for being eaten at the end of the Seder meal.

Syrian Jews have a custom of starting the storytelling of the Haggadah by taking the matzah used during the Seder, placing it into a special bag resembling a knapsack, and throwing it over their shoulders. They then proceed to recite the verse “מִשְׁאֲרֹתָ֛ם צְרֻרֹ֥ת בְּשִׂמְלֹתָ֖ם עַל־שִׁכְמָֽם

Each person takes a turn placing the bag of matzah over their shoulder. They are asked questions in Arabic which they respond to in Hebrew:

MinWen Jaye? Where are you coming from?

MiMiṣrayim -- From Egypt!

LaWeyin Rayeḥ? Where are you going?

LiYerushalayim -- To Jerusalem!

Ishu zawatak? What are you carrying?

Matzah u Maror -- Matzah and Bitter Herbs

This moment allows each person to place themselves back in time and replay a scene: The Jews are wandering the desert, they come upon some locals. The locals all speak Arabic but the Jews are speaking Hebrew, because they are free.

Approach #3: we eat Matzot, as it is Bread of Affliction

לֹא־תֹאכַ֤ל עָלָיו֙ חָמֵ֔ץ שִׁבְעַ֥ת יָמִ֛ים תֹּֽאכַל־עָלָ֥יו מַצּ֖וֹת לֶ֣חֶם עֹ֑נִי כִּ֣י בְחִפָּז֗וֹן יָצָ֙אתָ֙ מֵאֶ֣רֶץ מִצְרַ֔יִם לְמַ֣עַן תִּזְכֹּר֔ אֶת־י֤וֹם צֵֽאתְךָ֙ מֵאֶ֣רֶץ מִצְרַ֔יִם כֹּ֖ל יְמֵ֥י חַיֶּֽיךָ׃

You shall not eat anything leavened with it; for seven days thereafter you shall eat unleavened bread, bread of distress—for you departed from the land of Egypt hurriedly—so that you may remember the day of your departure from the land of Egypt as long as you live.
לחם עני. לֶחֶם שֶׁמַּזְכִּיר אֶת הָעֹנִי שֶׁנִּתְעַנּוּ בְּמִצְרַיִם (ספרי):
לחם עני THE BREAD OF AFFLICTION — i.e. bread that calls to mind the affliction to which they were subjected in Egypt (Sifrei Devarim 130:5).

כי בחפזון יצאת. וְלֹא הִסְפִּיק בָּצֵק לְהַחֲמִיץ וְזֶה יִהְיֶה לְךָ לְזִכָּרוֹן; וְחִפָּזוֹן לֹא שֶׁלְּךָ הָיָה אֶלָּא שֶׁל מִצְרַיִם, שֶׁכֵּן הוּא אוֹמֵר "וַתֶּחֱזַק מִצְרַיִם עַל הָעָם וְגוֹ'" (ספרי):

כי בחפזון יצאת FOR THOU CAMEST FORTH [FROM THE LAND OF EGYPT] IN HASTE, and the dough therefore had no time to become leavened, and this (the eating of unleavened bread) shall be unto you as reminder of this. The haste, spoken of here, was not on thy part, but on Egypt’s part, for so it states, (Exodus 12:33) “And Egypt was urgent upon the people [hastening to send them out of the land]” (Sifrei Devarim 130:6; cf. Berakhot 9a).

לחם עני. פרט לעיסת שנלושה ביין ושמן ודבש דלאו לחם עני היא .
(פסחים ל"ז א')

Other important things we learn from Shemot Chapter 12

Pesach night as 'לֵ֣יל שִׁמֻּרִ֥ים'

לֵ֣יל שִׁמֻּרִ֥ים הוּא֙ לַֽה' לְהוֹצִיאָ֖ם מֵאֶ֣רֶץ מִצְרָ֑יִם הֽוּא־הַלַּ֤יְלָה הַזֶּה֙ לַֽה' שִׁמֻּרִ֛ים לְכָל־בְּנֵ֥י יִשְׂרָאֵ֖ל לְדֹרֹתָֽם׃ (פ)

That was for ה' a night of vigil to bring them out of the land of Egypt; that same night is יהוה’s, one of vigil for all the children of Israel throughout the ages.

שמורים לכל בני ישראל לדורותם משומר ובא מן המזיקין. הודיענו בזה שהשמו' הזה הוא מן המזיקין ואמר משומר ובא כלומר שהשמור הזה הוא נמשך ובא בכל דור ודור דמדכתיב שמורים בלשון רבים ולא שמור בלשון יחיד הורה שיהיה שמור אחר שמור בכל דור ודור ולא שהשמור הזה יהיה בעבור כל דור ודור בלילה הזה:

Pesach night centered around the Home

בְּבַ֤יִת אֶחָד֙ יֵאָכֵ֔ל לֹא־תוֹצִ֧יא מִן־הַבַּ֛יִת מִן־הַבָּשָׂ֖ר ח֑וּצָה וְעֶ֖צֶם לֹ֥א תִשְׁבְּרוּ־בֽוֹ׃

It shall be eaten in one house: you shall not take any of the flesh outside the house; nor shall you break a bone of it.

תָּנוּ רַבָּנַן: מִצְוַת חֲנוּכָּה, נֵר אִישׁ וּבֵיתוֹ.

The Sages taught in a baraita: The basic mitzva of Hanukkah is each day to have a light kindled by a person, the head of the household, for himself and his household. And the mehadrin, i.e., those who are meticulous in the performance of mitzvot, kindle a light for each and every one in the household. And the mehadrin min hamehadrin, who are even more meticulous, adjust the number of lights daily. Beit Shammai and Beit Hillel disagree as to the nature of that adjustment. Beit Shammai say: On the first day one kindles eight lights and, from there on, gradually decreases the number of lights until, on the last day of Hanukkah, he kindles one light. And Beit Hillel say: On the first day one kindles one light, and from there on, gradually increases the number of lights until, on the last day, he kindles eight lights.

It is probably not coincidental that the most popular Jewish holidays, Pesach and Chanukah, are both centered around the home

Shabbat Hagadol

(ג) דַּבְּר֗וּ אֶֽל־כׇּל־עֲדַ֤ת יִשְׂרָאֵל֙ לֵאמֹ֔ר בֶּעָשֹׂ֖ר לַחֹ֣דֶשׁ הַזֶּ֑ה וְיִקְח֣וּ לָהֶ֗ם אִ֛ישׁ שֶׂ֥ה לְבֵית־אָבֹ֖ת שֶׂ֥ה לַבָּֽיִת׃

(3) Speak to the community leadership of Israel*community leadership of Israel Heb. kol ‘adat yisra’el, lit. “whole community of Israel,” which here denotes the part (“leadership”) that acts on behalf of the whole (“community”). See the Dictionary under ‘edah. and say that on the tenth of this month each of them shall take a lamb*lamb Or “kid.” Heb. seh means either “sheep” or “goat”; cf. v. 5. to a family, a lamb to a household.

(א) בעשר לחודש הזה ויקחו להם יום שבת היה ולפי שעשו בו ישראל מצוה ראשונה כגון מקח הפסח בעשור נקרא שבת הגדול.

(1) בעשור לחודש הזה, ויקחו להם, “on the tenth of this month, when they shall take or purchase for themselves etc.” The tenth of the month during that year was on a Sabbath. As a reminder of the fact that on that Sabbath the people collectively performed this first commandment, it has since been named שבת הגדול, “the great Sabbath.” [Compare an article on this subject in the miluim to Torah shleymah by Rabbi Menachem Kasher of blessed memory Ed.]
The Yom Tov law of 'אוכל נפש'

וּבַיּ֤וֹם הָרִאשׁוֹן֙ מִקְרָא־קֹ֔דֶשׁ וּבַיּוֹם֙ הַשְּׁבִיעִ֔י מִקְרָא־קֹ֖דֶשׁ יִהְיֶ֣ה לָכֶ֑ם כָּל־מְלָאכָה֙ לֹא־יֵעָשֶׂ֣ה בָהֶ֔ם אַ֚ךְ אֲשֶׁ֣ר יֵאָכֵ֣ל לְכָל־נֶ֔פֶשׁ ה֥וּא לְבַדּ֖וֹ יֵעָשֶׂ֥ה לָכֶֽם׃

You shall celebrate a sacred occasion on the first day, and a sacred occasion on the seventh day; no work at all shall be done on them; only what every person is to eat, that alone may be prepared for you.

אך אשר יאכל לכל נפש על שם שכתב כאן כל מלאכה, הוציא מכללה הדבר הנאכל ולכך כתיב בכל החגים במקום אחר כל מלאכת עבודה לא תעשו למעוטי הכנת אוכל נפש שאינה עבודה. אבל בשבת וביום הכפורים שאפילו הכנת אוכל נפש אסורה כתיב בהן כל מלאכה לא תעשו.

אך אשר יאכל לכל נפש, “However (work needed) for preparing the food for the person is permitted to be performed also on those two days. The reason why the Torah had to add this line is because it had written previously that כל מלאכה, every manner of work is prohibited on these two days, it had to spell out the exception to that rule. On another occasion (Leviticus 23,7) where the festivals are discussed, the work prohibition was mentioned already with the term מלאכת עבודה, the type of menial work performed on the week days, instead of the Torah writing: כל מלאכה “all manner of work.” The Torah wishes us to know that preparing food is not considered “menial work.” The only days on which even this kind of work is forbidden are the Sabbath and Yom Kippur, (which is also called Sabbath) [and had been forbidden in the desert before the Torah had been given as soon as the manna had fallen on the sixth day. Ed.]