R' Samson Raphael Hirsch construes the commandments of both Rosh Chodesh and the Passover offering as means toward fulfilling God's promise/commandment in Exodus 6:7:
... וְלָקַחְתִּי אֶתְכֶם לִי לְעָם, וְהָיִיתִי לָכֶם לֵאלֹהִים
and I will take you to Me for a people, and I will be to you a God ...
According to R' Hirsch (in his commentary on 6:7 and then on 12:3-13), God designed the Passover offering experience as the process that would transform the crowd of dehumanized, dispossessed slaves that was the Israelites into a nation - His nation. I'll summarize some of the relevant points here, but as usual, I recommend reading through his entire magnificent treatment of this portion yourself.
... וְיִקְחוּ לָהֶם, אִישׁ שֶׂה ...
... they shall take to them every man a lamb ...
Each individual gained personhood by acquiring personal property and by becoming eligible to be an agent for others.
שֶׂה לְבֵית-אָבֹת--שֶׂה לַבָּיִת ...
... a lamb, according to their fathers' houses, a lamb for a household;
God gathers the individuals first into families and extended families.
וְאִם-יִמְעַט הַבַּיִת, מִהְיוֹת מִשֶּׂה--וְלָקַח הוּא וּשְׁכֵנוֹ הַקָּרֹב אֶל-בֵּיתוֹ, בְּמִכְסַת נְפָשֹׁת: אִישׁ לְפִי אָכְלוֹ, תָּכֹסּוּ עַל-הַשֶּׂה.
and if the household be too little for a lamb, then shall he and his neighbour next unto his house take one according to the number of the souls; according to every man's eating ye shall make your count for the lamb.
What cements a bunch of agency-endowed individuals into a Jewish nation is mutual aid. If one person has more than he needs, he contributes the extra to help someone else who doesn't have what he needs.
The lamb is always associated with the shepherd leading it around. When the Israelites offer one, representing themselves, they're submitting to God's guidance and leadership. A whole nation doing this becomes God's nation.
God took the Israelites to be His nation via this initial Passover offering. He will be to us a God throughout the generations through regular relationship-renewing meetings between God and nation. This section of the Torah describes both monthly and annual meetings.
First, in 12:2:
הַחֹדֶשׁ הַזֶּה לָכֶם, רֹאשׁ חֳדָשִׁים: רִאשׁוֹן הוּא לָכֶם, לְחָדְשֵׁי הַשָּׁנָה.
This month shall be unto you the beginning of months; it shall be the first month of the year to you.
R' Hirsch notes the significance of the word "לָכֶם" - "unto you (pl.)" here, in light of the rules we have for inaugurating a Rosh Chodesh. Each monthly meeting with God is established not by God, not by astronomical phenomena, not by individual Jews, but by the Jewish nation as a whole, through its leadership - the Beit Din.
R' Hirsch translates "חֹדֶשׁ" not as "month" but as "renewal of the moon," which serves as a reminder for us to renew our relationship with God. As a result of these periodic rejuvenations, we won't slide into the sort of absolutely obstinate God-ignorance that was cultivated in the stagnant culture of Pharaoh's Egypt.
Similarly, the annual holiday of Passover (commanded in 12:14-20) brings us back each year to the original moment of God's taking us to be His nation and invites us to re-commit ourselves to His shepherding. Like the command for Rosh Chodesh, this command (in 12:16) uses the word "לָכֶם" - "unto you (pl.)" to put the responsibility for this annual re-connection on the nation as a whole. Now that God has taken us to be his nation, we and He can work together, month by month, year by year, to forever ensure that He is our God.