Summarized from "Berosh Hashanah Yikatevun" (Michlelet Hertzog, edited by R' Bazak)
The calendar and the new year (R' Yoel Bin Nun)
RH is not quite the "beginning of the year" as a literal translation would indicate. From Tanach it is clear that the entire period between RH and Shemini Atzeret is the "beginning of the year". RH is simply the "beginning of the beginning of the year", the first day of the period which marks the transition between years. Thus, Yovel is announced on Yom Kippur, which is not after the beginning of the year, but rather in the middle of the beginning of the year.
We can also explain why Yom Kippur is on the 10th day of the month, unlike any other holiday. In a non-leap-year, Yom Kippur falls 364 days, plus or minus one, after the previous RH. Thus, we are being judged, almost exactly, for what we have done over the previous solar year - from last RH until this Yom Kippur. RH and YK are different dates from each other, but they fall within the same *period* of transition between years.
Reenactment (R' Yoni Grossman)
Pesach is obviously a holiday in which we not only remember, but are required to reenact and see ourselves as having left Egypt. Other holidays also require us to "pretend". Chanukah candles have several laws which are taken from the laws of the Temple menorah, implying that our lighting is a reenactment of the Temple lighting. And Yom Kippur mussaf includes a reenactment of that day's service, in which we bow down just when people in the Temple would.
The same may be true of RH. In your thoughts and prayers, you experience the creation of the world. That is to say, you feel what it means to have creation from nothingness, and are aware of the fact that you are such a creation.
The Rosh Hashanah offering (R' Avraham Walfish, plus a little from myself)
RH is the only holiday on which the main mitzvah of the day is not related to the Temple. (This is of course connected to the fact that it is also probably the most universal holiday, and the one least centered on the Jewish people.)
On the three agricultural holidays (Pesach, Shavuot, and Sukkot), special offerings were brought in the Temple (omer/barley, bikurim/fruit, water libations, respectively) as recognition of the fact that Israel was about to be judged regarding that season's agricultural aspect (grain harvest, fruit harvest, rain). In each case, what you offered as a sacrifice was the same product which you hope to receive.
On RH, judgment is not passed on a particular agricultural product. Instead, general judgment is passed on humans as individuals. Therefore, the "offering" which we need to bring consists of ourselves - that is to say, our attitude during the prayers, reflecting our new and improved moral qualities. We have been working on these aspects of our personality during Elul, and now it is time to present them to God.
The mitzvah of the shofar is unusual in that we are required not to blow the shofar, but to hear its blowing. (Granted, there is controversy about this in the sources, but certainly there is some independent value to the hearing.) The mitzvah thus focuses on your experience of the shofar, and not simply on the physical action. Apparently this is because the shofar's main purpose is to affect us emotionally. It helps us achieve the emotional state which we can then "offer" to God as a "sacrifice".
Kiddush hachodesh (R' Yehudah Shaviv)
The mishna in RH starts off with two chapters on kiddush hachodesh and only then discusses the shofar and the RH prayers. This is the case because kiddush hachodesh is actually, functionally, a crucial mitzvah on RH. When kiddush hachodesh was not automatic, everyone would be staring up at the sky on the night of RH to find out if they were really celebrating RH or if it was just a weekday after all. The "night" mitzvah of RH was kiddush hachodesh, and the "day" mitzvah was shofar, so it makes sense that kiddush hachodesh is listed first.