You can see the more interesting parts of Tokyo by using the Yamanote train line, which runs in a circle around the heart of the city. On the Yamanote line near Tokyo station you'll find Ginza, with it's expensive shops (and used camera stores).
A couple of stops up the line is Akihabara, which is the old "Electric Town" where you'll find lots of computer/game/souvenir/electronics shops and duty-free shops. Be warned, you won't find any bargains here, if you are from America, you'll find better prices on Japanese goods there than you will find in Japan.
A couple more stops up the line is Ueno, which is famous for it's museums, park, zoo, and shrines. If you want a look at Japanese history, the Tokyo National Museum is in Ueno. The museums and zoo are closed on the first Monday of every month.
Further along the line you'll come to Shinjuku, which is home to Kabukicho, and probably one of the best places you'll find in Tokyo for street photography. You'll see a lot of "interesting" people in the area, "hosts" who are fantastically dressed young men who try to hustle women into their clubs and then empty their purses by talking them into buying expensive drinks, or hustlers with offers like "sex with a Japanese girl...". It's an interesting place to explore, with lots of small streets and alleys.
The next place along the line is Harajuku, which is best to visit on Sundays when the young cosplayers hang out on the bridge by the station showing off their "fashion". You'll know you are in the right place if you see lots of foreigners and Japanese shooting cameras at young people dressed like super-heros or comic book characters. Near Harajuku is Yoyogi Park, which is home to Meiji Jingu shrine, which is also worth the time to see. Across from Harajuku station you'll find Harajuku street, and then Omotesando (where I live). Harajuku street is full of small boutique shops frequented by young people, whereas Omotesando is full of high-end brand-name stores (bring your platinum card).
Next stop is Shibuya, famous in "Lost in Translation", or "Tokyo Drift". Shibuya is one of the not-to-be-missed parts of Tokyo. Look for the Hachiko exit, and make sure to pose for a picture with the the bronze statue of the dog Hachi (which inspired the recent Richard Gere movie "Hachi"). You'll find countless more shops in Shibuya, and lots of good restaurants.
If you stay on the Yamanote train, it will eventually take you back to Tokyo Station.
Another good place to visit is Asakusa. Near Tokyo Station you'll find Takaracho Station, or Kiyobashi Station, where you can take either the Asakusa Line or Ginza Line subways to Asakusa Station. Asakusa is home to the Sensoji temple, and the largest assortment of souvenir shops in Tokyo. The few remaining "old town" parts of Tokyo are located in the Asakusa area, so it is worth seeing.
One weekend is not really enough to see all there is to see. I've lived in Tokyo for three years, and I have not yet come close to seeing it all.
