It is tradition in Japan to celebrate the new year at a Shinto shrine (hatsumode). The protagonists will offer some money, ring a bell at the end of a thick rope, and then pray for good fortune.
This usually goes along with gorging yourself at the many temporary food stalls, playing various games, and drawing a fortune.
The female characters in the anime will usually dress up in a kimono, to show off their traditional Japanese clothing, while watching the fireworks.