It looks like you've got plenty of advice already (haven't actually bothered to read all 7 pages) but as someone who has been 3 times and knows the Kansai region and Tokyo fairly well I feel qualified to chime in
When you go is important - will this be a summer or autumn trip? If you're going in summer it'll be real hot, but that means you can enjoy the summer festivals, seasonal dishes (crushed ice if nothing else) and the deafening sound of the cicadas. The best festivals around Kyoto to look out for are Gion Matsuri and the Daimonji festival, where giant kanji are set alight in the hills surrounding the city. Plan your trip accordingly.
If you're planning an Autumn trip then November is a great time to go. Kyoto looks beautiful in the Autumn and it's a whole lot more pleasant to walk around than during the heat of say, July
You've probably figured out that I like Kyoto (the whole Kansai region really) an awful lot - so much so that I actually don't think it's unreasonable to tell first time travellers not to bother with Tokyo at all. If you want a sprawling metropolis Tokyo is pretty much unbeatable in the world (impossibly clean, totally safe, full of interesting things to do) but anecdotally I've found many friends (even my own sister who's a London girl) find it a bit overwhelming. By contrast, I've yet to meet anyone that hasn't loved Kyoto. Plus, if you want more of a hustle and bustle, then Osaka and Kobe are only short train trips from Kyoto
If however you're planning on doing both Kyoto and Tokyo then get yourself a JR Pass and abuse the hell out of it. Bullet train yourself from Tokyo to Kyoto to Nagoya to Osaka to wherever your heart desires - if you're spending a few hundred on a train ticket you should get your money's worth
Other trips outside Tokyo/Kyoto:
- Nikko and Kamakura; not essential if you're doing Kyoto and Nara, but pleasant places. Both short trips from Tokyo.
- Hakone; probably one of the better places for onsen in the country, and it's en route from Tokyo to Kyoto. Onsen is a must-do if you're travelling in the cooler months.
- Nara; unmissable. Short trip from Kyoto. Say hello to the giant bronze Buddha and the thousands of friendly shika dear. Just don't leave any food in your back pockets.
- Kobe; imagine Japanese San Francisco. Good beef too
If you're staying in Tokyo for a while I cannot recommend
this place enough. Very well priced and well located in Ikebukuro which is only a short ride from Shinjuku, Shibuya, Harajuku and so on.
When in Kyoto, if you go drinking, you must go to
this bar. Probably the coolest bar I've ever visited, run by a French magician.
If you only go to one bar in Tokyo, I nominate the New York Bar, famous for being used on the set of Lost In Translation. Spectular view of Tokyo at night and live music. Expensive mind you, so get in before the cover charge (7:30pm iirc)
I could go on and on forever with things to do so I'll finish here for now, especially since it's possible others have said plenty already