I would honestly only recommend the Rail Pass if you plan on hitting multiple major cities via Shinkansen. Other than the train from the Airport (assuming you take the N'EX from Narita into Tokyo as most do), most of the trains in and around the city are pretty cheap. Even if you account for 4000 yen each way on the N'EX (that's what the fare was last time I went I think), it would take a lot of local trains to add up to the cost of the rail pass. One round trip on the Shinkansen from Tokyo to Osaka and back, though, and that rail pass is mostly paid off.
While I recommend getting out of Tokyo and into "real" Japan, a lot of first-timers or travelers that don't know much Japanese can still do a lot in and around Tokyo for 1-2 weeks sticking to local trains.
As for my recommended go-to spots:
Akihabara (Tokyo) and Den Den Town (Osaka) of course.
Nakano (short trip from Tokyo Station on the Chuo line).
Kokubunji (A little further down the Chuo Line than Nakano. Small urban area with some great game centers and independent game shops).
Sannomiya (A really great shopping and entertainment area in Kyoto)
Ootsu (A small lakeside resort town not far from Kyoto).
Also -
Hyperdia is a great tool for those moments you find or hear about something new to do but don't know how to get there. Plug in the names of the nearest train stations, and Hyperdia will show you all the ways you can get there - What trains, their schedules, any transfers (and how much time you'll have to make them), and the fares. It's an absolute godsend whether you are just visiting or are living there long term.