There is an argument about when to celebrate the festivals. Generally, people go by the calendar introduced by Meton in Athens. This got accepted during the middle ages as being ''halacha leMoshe  miSinai'' [given to Moses at Sinai.] [note 1] However it is not mentioned in the Gemara [Talmud]. My approach is go by Tosphot in Sanhedrin page 10 side b where the molad [conjunction of the moon and sun ] is considered as the start of the new month. 

[When the Talmud says that nowadays we know the time of the new moon in several places it does not say anything about the calendar. It just says we know the time. nor is Hillel II mentioned in the Gemara in connection with any calendar.]] 

So that the day of the molad is rosh hashanah. And the best way to celebrate rosh hashanh is to learn Torah. [i.e. to spend the day learning Gemara, Tosphot, Maharsha, and the Avi Ezri of Rav Shach]

[note 1] In the responsa of the geonim there are plenty of dates which do not correspond to the calendar--thus showing that the calendar was accepted later.