I took a trip to Japan a few years ago with some friends and we visited Tokyo, Kyoto and Osaka. We spent about 2 weeks in Tokyo (first and last week), 4 days in Kyoto and 3 days in Osaka. First off, I would recommend skipping Osaka and sticking to Tokyo and Kyoto, maybe a week at each place.

For travel between Tokyo and Kyoto, I agree with the others and recommend the Shinkansen. I think it cost us about $130 a person each way for the fast N700. It's something you need to experience and don't forget to watch for Mt. Fuji. Everyone in the group thought it was well worth the expense.

Osaka was difficult to get around and find things to do. The Osaka Castle grounds were neat but the actual castle is a mockup with a museum inside, a bit of a letdown. Dotonbori was neat (most people compare it to city scenes in Blade Runner) and there are a lot of neat food areas but I felt it was lacking the Japanese culture I wanted to see as a tourist. It seems like it was only about 15 minutes from Kyoto so you may just want to go there for a day trip. The aquarium was neat.

Tokyo was the easiest to get around. Pick up a Pasmo (? been a while) card for getting around town. Everything in the subway is written and announced in English so you should have no problem getting around. Many restaurants will also have English menus (if you are ordering at a machine check on top for an English Language menu). I highly recommend visiting Akihabara (the electronics district), Ueno Park (Koen) and the market alley by the subway stop, the Imperial Palace and surrounding grounds, Ropongi at night (prepare for a hard sale by the street hawkers), Harajuku bridge for the cosplay on a Sunday afternoon, Shibuya for the busy intersection and fascinating stores behind the Starbucks, Senso Ji temple in Asakusa and for a view of the city skip Tokyo Tower and go to the Government Buildings in Shinjuku (free to go up to the 43rd floor in either tower and a much better view). I think they shut down access to the Tsukiji fish market but if not it is fascinating place to wander around. I know you said no hostels but we stayed at K's House Tokyo and the facilities were great. They scrubbed that place from top to bottom daily and it looks like they have a private 2 person ensuite (bathroom in the room) for 9800yen a night (about $125). It was located in the Asakusa district close to several subway stops, an easy walk to Sensoji and right off the Sumida River. This was my first hostel and I was very impressed.

To me Kyoto was the real gem of my travels. It embodied everything I wanted to see in Japan. Just to the northwest (west?) of town is Arashiyama. Go there and visit Tenryuji (by the way, ji is temple and jo is castle). There is a rock garden in the front, you can tour the building (japanese screens and tatami mats), a japanese garden with pond in the back and wander up the hill to see the bamboo forest with bamboo as thick as your arm. Be sure to take a full day in Arashiyama and wander around to several of the other temples and down to the river. There are dozens in the area. Back in Kyoto, I recommend visiting Nijojo. It's great to wander around the grounds and walk through the building and hear about the nightingale floors and hidden spaces. I also highly recommend going to see the Golden Palace. I wish I had more time in Kyoto to see more. This is also the place to check out some Izakayas (red lantern bars). Just make sure to leave if they say no gaijin. As for a place to stay, they have a K's House Hostel here too. Once again it looks like they have a 2 person ensuite but the rate of 3900-4400yen/night sounds a little low so I am guessing it is per person. It's a quick walk from the train station (maybe 10 minutes). When we were there, they were building a new facility right next door. Once again, these are super clean facilities and the common room is a great way to get info on what to do while you are there. Travel in Kyoto was a little more difficult. We ended up using a mixture of bus, subway and taxi.

I'm envious of your trip.