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The minhag to eat special fruits in Rosh HaShanah is from the Gemara in Horiyos 12a. While the gemara gives a list of recommended fruits dipping apples in honey is not mentioned.
The earliest source for the apple in honey is from the Abudraham as quoted by the Rema (OC 583:1). What is the significance of this famous minhag? The Moadim L’Simcha brings a number of reasons.
1. The Maharil says that Yitzchok smelled an apple orchard when he commented on the beautiful smell as Yaakov came in to him to receive the Brachos. The Vilna Gaon in the Biur HaGra says that this happened on Rosh HaShanah.

This list was compiled by Rav Saadia Gaon:
1. The Shofar is like the trumpet which announces the coronation of a king. That is why it is used on Rosh Hashanah, the birthday of the universe. At that time we accept Hashem’s Rulership. Our tefilos and shofar blasts are like the coronation ceremony in which Klal Yisroel crowns Hashem as Sovereign.
2. The Shofar calls us to examine our deeds and return to Hashem, who will always accept us if we are sincere. This is why we blow the Shofar on Rosh Hashana which is the first of the Aseres Yimei Tshuva.
3. The Shofar reminds us of the Shofar which blew when the Torah was given at Har Sinai; thus we are reminded to study and cherish Hashem’s Word.

By Rabbi Doniel Neustadt
One of the most important mitzvos of Rosh Hashanah is the Biblical command to blow the shofar.

We all know that Rosh HaShana is not a day where we bombard Hashem with our wish list for the upcoming year. But is it a day of Tshuva? On one hand we don’t say Slichos and avoid all mention of any reference to aveiros on this day, but on the other hand we blow the Shofar which is very much tied to tshuva. We also know that there are Aseres Yimei Tshuva of which Rosh HaShana is part of. So is it a day of tshuva or is it just a day to coronate our King and kick off our new year?

By Rabbi Y. Dov Krakowski
The following is meant as a convenient review of Halachos pertaining to Rosh Hashana. The Piskei Din for the most part are based purely on the Sugyos, Shulchan Aruch and Ramah, and the Mishna Berura, unless stated otherwise. They are based on my understanding of the aforementioned texts through the teachings of my Rebeim. As individual circumstances are often important in determining the psak in specific cases, and as there may be different approaches to some of the issues, one should always check with one’s Rov first.
Hilchos Uminhagei Rosh Hashana:
 
Erev Rosh Hashana: