Rosh Hashanah is the first and second days of the first Jewish month of Tishrei. It marks the beginning of the Jewish new year. The celebration of this holiday is marked with solemnity, as it is the day on which the whole world is judged for the coming year. Rosh Hashanah is the birthday of the world, as it was on this day that G-d created Man on the 6th day of creation. Every year, on this day, we proclaim G-d as our one and true King.
The Talmud relates that Man was created on the first of Tishrei. This being the case, Rosh Hashanah is a birthday of sorts for the human race.
On Rosh Hashanah we celebrate our humanity by exercising our free will.
Rosh Hashanah is a happy day, a festival, and at the same time it is a very serious day. It is a serious day because it is the day of judgment, and it is a happy day because we are confident that if we understand the meaning of the day and use it properly, then we will indeed receive a favorable judgment.

We dip an apple in honey because it tastes good! Also, because it represents our heartfelt wishes for a sweet year, not only for ourselves and our families but also for all the Jewish people.

We blow the shofar since Rosh Hashanah is the anniversary of the creation of the world, it follows that it is also the anniversary of God being sovereign over the world. Rosh Hashanah is a coronation of sorts, and thus we trumpet the shofar just like at a coronation ceremony.
Source : http://www.jewfaq.org/holiday5.htm