Why don’t we begin reading the Torah again from Bereshit (Genesis) on Rosh Hashana?

That’s a great question. Rosh Hashana is the anniversary of creation. Even though we know that the Torah was created before the world, it was not actually given to the Jewish people until Mount Sinai. Shavuot is the anniversary of receiving the Torah, but we celebrate the Oral Law then (there are many reasons why it is the festival of the Oral Law – go and investigate). The Written Torah was really received on Yom Kippur, when Moshe came back down Mount Sinai for the 3rd time with the second set of tablets. It is this event that begins the Torah portion of Yitro (Exodus 18; 13 and what follows) which is the prelude to the Ten Commandments. (Commentary of Rashi explains that the ‘next day’ refers to the day after Yom Kippur). However, it would be inappropriate to celebrate the giving of the Torah on Yom Kippur, since that is the day of repentance and forgiveness.. So we wait until straight afterwards, with Succot, and Simchat Torah, (which is in fact the culmination of Yom Kippur) and begin reading the Torah again on that date.

Wishing you a happy and successful year,
Rabbi David Sedley