Take the trains, it's the best way to get around. Here's a protip: find a bookstore somewhere and see if they sell an English booklet to getting around (should be a bunch). I picked up an excellent one several years ago which was invaluable in deciphering the myriad of train stations and schedules.
As far as places to see, I'm kind of bored with Tokyo as there are only so many touristy places to see. I personally hate tours so did a lot more research to find places where I wanted to see on my own schedule. So without further ado,
Tsukiji (fish market)
Hamarikyu (this is close to the fish market and you can take the ferry to Asakusa from here)
Asakusa (not that far away and a short ferry ride away if you want)
Ginza
Imperial Palace
Harajuku
Meiji Temple (close to Harajuku)
Akihabara (electronics)
Roppongi
Otherwise, just walk around, most people are pretty polite and more than happy to help get you to where you need to go. If you like ramen (not that instant noodle crap), definitely make an effort to try different places. Or check out the Ramen Museum
Hakone is not that far away if you have a little extra time and want to check out onsen (the baths)