You will be deluged in Japanese culture no matter what you do.
Spring and fall are the best times to visit Japan. Cost-wise, it is expensive, especially with the current economy.
However, I recently spent 4 nights in Kyoto staying at two different hostels that were each less than $30 per person per night.
Hostels.com served me well.
Tokyo will be more expensive than Kyoto. Both have plenty of shrines and temples and these can become boring quickly. Kyoto would be better for scenery, Tokyo better for nightlife.
Obviously there are lots of other places, but these are my personal experiences.
As for getting there, that obviously depends where you are coming from.
japan-guide.com is useful.
This may also help:
http://www.japan-guide.com/e/e2362.html
It's a train ticket that gives 5 days of unlimited travel on ordinary trains (not the bullet train) for 11500 yen (about $130). The downside is it's only available during certain times of the year. It's also a very time consuming means of travel, since it doesn't use the bullet train, and the transfers could be confusing if you don't know Japanese.
Japan Railways also has excellent ticket packages for tourists. However, these must be purchased outside the country.
Hope that gets you started.
參考: I'm studying for a year in Tokyo right now.