The book of Matthew starts with the geneology of Jesus because this was very important to the Jews. The Jewish OT prediced that the Messiah would come from the line of David, so it was very important to show how Jesus fulfilled this prophecy. This is the first of the 4 gospels and focuses heavily on how Jesus fulfilled the Jewish prophecies.
The Jews did not believe it was proper to mention the name of God, so in this gospel, Matthew observes the Jewish tradition of using the phrase "Kingdom of Heaven" instead of "Kingdom of God."
Matthew’s goal was to show that Jesus was the promised Messiah, the one that Moses spoke about in the law and the one of whom the Jewish prophets wrote about. There are many allusions to the Old Testament in this book and many, many more direct quotations from the Jewish Old Testament (the books of Moses) than any other book in the New Testament. Matthew tries to show that Jesus came NOT to destroy the Jewish law, but to fulfill it. Jesus is presented as the Messiah, the heir to the throne of King David, the rightful King of the Jews. The genealogy, the story of the wise men, the killing of the innocents and the flight into Egypt were all meant establish Jesus as the predicted Messiah of Israel.
The Jews did not and do not accept Jesus as Messiah. They beleive Messiah is still to come. Jesus is the “stone that the builders rejected.” The Jews at the time were hoping Messiah would come and free them from Roman rule. They expected a savior on a white horse, which is also predicted in the Old Testament. Christians believe that Jesus will come on a white horse also, but it will be the second time he comes. Jesus did not meet the expectations of the Jews when he came as a humble servant and was born in a stable.