First, get this book, as the author includes the phonetic pronunciations (muy importante)
Linky wa koko desu!
How are you travelling? Check into a week long Railpass and whether it includes the Shinkansen (believe some do and some don't). The Railpass can only be purchased OUTSIDE Japan. Bullet train from Kyoto/Osaka to Tokyo about two and half to three and half hours (depends on what kind of Shinkansen). For metropolitan areas, trains are the best way to go.
Write to (or E-mail) Japanese Imperial Household Agency to get passes to see Imperial Palace in Kyoto. The pass must be requested one month prior (for foreigners may be two weeks). A must see, however, may skip Imperial Summer Palace outside Kyoto. Too much other stuff to list.
About the only thing I thought worth the time in Osaka is Osaka-jo (Osaka castle). But, you may want to visit Universal Studios (just opened last year) close to Osaka. Can easily reach Osaka-jo by train, tour and return to Kyoto in a morning, so may be better to stay Kyoto three nights and then up to Tokyo.
For just walking around to soak up Tokyo atmosphere, Shibuya would be better than Shinjuku. Shibuya, Shinjuku, Tokyo and Akihabara all easily reached on Yamanote-sen (Yamanote line) which is a train line that runs in a big circle in Tokyo.
PDF zoomable map of JR trains in Tokyo. Contrary to popular opinion, electronics are NOT drastically cheaper here. May save a few bucks on some items, but for computer stuff, newegg usually just as good or better. Same goes for cameras and camcorders.
FYI, have been to Kyoto twice, Osaka twice and go to Akihabara about once a month.
Have some pics can send (have to change size first). PM if interested.