Needless to say, you've made a great decision to visit Japan, it is an amazing, very inspiring country. There are countless reasons to go, and the ones I would most like highlight are nature and culture. Japanese nature is really beautiful, I recommend visiting national parks away from the cities, of which there are plenty, and many Japanese-style gardens. About culture, I don't really need to explain...
I see you've chosen Tokyo and Osaka as primary destinations. In Tokyo, aside from things you've mentioned, I would most recommend going to the Asakusa shrine (and the lively markets around) and the SEGA Joypolis in Odaiba. I also highly recommend the notably cheap Utahiroba karaoke (especially cheap during the day in weekdays). There are a number of those around Tokyo (and elsewhere in the country). Around Tokyo, I highly recommend visiting Nikko and Kamakura. Well, Nikko is some 100km away, but really worth it. It's a wonderful town in the mountains with amazing shrines and temples in a forested area. Additionally, Ieyasu Tokugawa's grave is there, if I remember correctly. Kamakura has that gigantic Buddha statue, also several other beautiful temples, mountain hike trails, and even beaches!
In Osaka, I highly recommend Shinsekai. Very interesting neighbourhood. Also, Temma! The shrine is cool, but what's best is the countless second hand book shops in the arcades around it. I bought manga for dirt cheap there, there were many at 50 yen a piece (that's less than 50 US cents). Around Osaka, Kyoto, Nara and Kobe are obvious recommendations. Kyoto has rich history and straight roads, very hard to get lost. :) Golden Pavillion, 1000 torii gates place (Fushimi Inari), Kiyomizudera, Nijo castle, among many others. Rivers and bridges... Also, there's a great rolling sushi place called Sushi no Musashi. Oh, and Arashiyama on the outskirts of the city, with the beautiful bamboo forest! Nara is a former capital located in a ravine, with a really huge shrine, and lots of deer running around everywhere! They let you pet and feed them, and they'll likely try to snatch your ice cream if you get too close. Kobe I haven't spent much time in, but it's a big, youthful city with lots of neon lights. And of course Kobe beef, which is quite expensive, however, but worth a single try. Himeji is also relatively close, and has a beautiful castle, one of the best in Japan.
Let me know if you'll be able to travel also to other areas, so I can recommend other places.
Also, congrats on studying Japanese, it's a very interesting language! Keep at it and in 2-3 years you'll be surprised at the level of fluency you've reached. :)
Thanks very much for the intensive guide and helpful suggestions. I like traveling off the beaten path to some degree. This gives us some idea of how to explore.
If we can get the time off, I'd prefer to spend two weeks at a crack and go to 3-4 primary destinations [Tokyo/Osaka/Kyoto] and take them each in for a few days without feeling rushed. Thanks again! We'll have a better idea of where we are specifically planning to go when we get closer to the time of our trip. ---
essentially shizuo "eat your vitamins and say your prayers" heiwajima in an apron.