I agree with most postings before mine. Would suggest Tokyo and Kyoto and skip Osaka because you will not miss much. Kyoto is no question the better option for temples and traditional japanese experience. For nightlife, you’ll have that covered in Tokyo. What I did was stop over Osaka for a few hour day trip, that should be plenty. You can leave and lock your backpack at the Osaka train station.

It all depends on how much you enjoy visiting temples/historical places. A friend of mine didn’t think highly of Kyoto because it’s boring to her, but I absolutely loved it because of the real “japanese” experience more than Tokyo because to me it’s just another big city.

But I also totally second MisterOL’s suggestion to visit Himeji if you can. Nara is close by as some others have suggested, take one day out to drop by Nara if you like Kyoto.