The Shinto Shrine is the only busy place you'll find on New Year's. Many people on the night of Dec. 31st go to a Buddhist temple to hear the 108 bongs of the bell, which are supposed to drive off the 108 sins of the human condition. Traditionally, many visit 3 different shrines on New Year's, and a few actually stay up all night to do it and see the first sunrise of the year. It's about the only time that public transportation is running in the wee hours of the morning; normally they shut down before midnight. Some of the more famous shrines will be jammed with people, as they go up to the altar and pray for health and prosperity for the coming year. It is also one of the few times that you can see some Japanese women in a traditional kimono. Japanese also receive Nenga-jo or New Year's Cards, and that is the only mail delivered on Jan. 1st. Some mail out several hundred cards to every acquaintance and business contact they have, much to the delight of the Post Office which makes godzillions of yen from it.
Fireworks are generally NOT used, but this too due to the whoop-it-up atmosphere of the West is slowly changing, and in some more populated areas or amusement parks you can see some. You won't find any home fireworks on sale anywhere though, and if you want to light some up you had better buy them when they are on sale, during the summer usually til the end of August.