Celebrate hatsumode (the first shrine/temple visit of the new year) at Zojoji Temple, with the illuminated Tokyo Tower in the background. Get there before midnight to hear joya no kane, the Buddhist act of ringing the temple's bell 108 times, each ring to rid of a worldly desire, which is kind of like a sin. Once the clock strikes midnight, get in line to donate a ¥5 coin and make your wish for the upcoming year. There is also jorurikai (浄焚会), when fortunes from the previous year’s hatsumode are burned. Don't forget to pick up new year memorabilia such as ema (a wooden sign for writing a wish).
We recommend combining hatsumode with hatsuhinode (first sunrise of the year) at Tokyo Tower; they're right next door.