Well Tokyo certainly has enough to entertain you for 10 days but it really depends on whether you're into city stuff or not.
If you are then high on my list would be:Odaiba
(man-made island in Tokyo Bay with shopping, big onsen, and entertainment galore).Asakusa
is a lovely area - it's the old part of the city with Sensoji Temple and Nakamise Dori (old style shopping street)Shibuya
- go at night and check out the Hachiko area just be ready for a swarm of people. It can be disorientating at first!Ebisu
is a favorite place of mine to chill out and wander round. On the East side of the station is Ebisu Garden Place - very upmarket with nice restaurants etc. On the West side is older style Tokyo and you can walk up to the area of Daikanyama - lots of small funky shops.Ueno
- has culture by the bucket load - museums, park, zoo etc.Kichijoji
- is an area to the West of the central Tokyo on the Inokashira Line from Shibuya. This is where you'll find Studio Ghibli (a famous anime museum but to call it a museum is kinda wrong as its a really magical place. If you do want to go to Studio Ghibli you have to book it in advance - you'll need to talk to a travel agent for this as you can't do it online. Tickets are usually fully booked at least a month ahead of time so you'd need to be booking this by September at the latest I would say. Also in Kichijoji is Inokashira Park which is lovely.
Still.... if you're not a city lover then you may get fed up of the scenery. If you don't want to jump on the bullet train there are several nice places only an hour or so away from Tokyo - check Nikko, Mount Takao, and Hakone.
Otherwise, if you do decide to travel to Kyoto and/or other areas (Kyoto is beautiful by the way and if you're into temples there are many) check out the Japan Rail Pass option offered by JR. This can save you a lot of money if you are traveling across the country. If you're only staying in Tokyo then don't bother but using the bullet train is expensive! The more you intend to travel the more the Rail Pass will save you.
Take a peek at my website if you like - I've tried to put info together that will help http://www.tokyotopia.com