"Japanese culture" is extremely broad - you need to find more about what you like. For example, its food, music, castles, arts, temples/shrines, nature, gardens, hiking, night life, museums, shopping, history, and much more.
No visit to Japan would be complete without going to a nice hot spring (onsen) at least once. They are all over the country. But a few famous places are Hakone, Kusatsu, Kinosaki, Kurokawa, Beppu and Yufuin.
Most first timers split their time between Tokyo (the capital and modern heart of the country) and Kyoto/Nara (the traditional soul of Japan). But don't feel like you have to go down the same old beaten trail. If you want, branch out to other places like Okinawa or Hokkaido if you like.
If you come during the summer, the whole southern half of the country is extremely muggy and humid - going to the mountains or Hokkaido would in fact be a great escape.
There are good rail passes for both regions and the whole country.
There are also dirt cheap air fares you can take advantage of.
It's up to you to do your research. You can also read some good advice for new tourists as well. Just remember, don't cram too much in - Japan will always be here.
I want you to visit Tokyo and Kyoto first.Why don’t you joining the tour like this?
Of course , you can see around there by yourself.Then, you can use the services.