Practically every shrine in Japan will host a New Year's event. It is perhaps the most important day in the Shinto calendar as it involves the cleansing of all of the previous year's impurities and the welcoming of a fresh new year. I'm always impressed by how quickly we Japanese disposed of the previous lunar calendar unlike our Chinese brethren and now celebrate New Year's on January 1st on the Gregorian calendar.
Your local city newspaper or the Japan Times has information on the major shrines. Many train stations also have pamplets with descriptions of the various shrines and how to get there. If you're in Tokyo, go to Meiji Jingu in Harajuku for perhaps the largest crowd in Japan. In Kyoto, hit Heian Jingu.
Each shrine will have its own array of chachka* (stuff to sell you) from amulets that protect your house, your car, or make sure your wishes will come true this year. So bring some cash as well.
* I misspelled this originally: it's either tchotchke or chachka. It's a yiddish word meaning junk.
Equipment to bring: an SLR or rangefinder with a lens range of 24mm to about 90mm will suffice for most purposes. The events do tend to be extremely crowded. You won't be able to easily use a tripod although a monopod might work. Film speed around 100-400 should suffice. Don't plan to take pretty pictures of the actual shrines, for that pick another day. But you can get good crowd pictures as well as maybe some shots of the priests at work. A large camera backpack may mean you will be smacking some poor obaachan in the face, so try to keep your outfit compact.
ps. The major events are usually at Shinto shrines (jinja神社) and not Buddhist temples (tera 寺) - although many larger temples also hold events (it's also a bit confusing since many of the major temples also have shrines on the same grounds; and vice versa). Shrines can be recognized by the torii(鳥居) or giant red gateway as well as the folded white ribbons (gohei御幣) dangling from thick ropes (shimenawaしめ縄) tied around trees. This site has good information on the various accoutrements of shrines: http://www.japan-guide.com/e/e2059.html