The Japan Rail Pass, which allows tourists unlimited rides on all JR trains (including bullet trains across the country), is one of the best ways to see lots of different cities without breaking the bank.
For non-Japanese residents, or tourists with more limited time or means, there is also the JR TOKYO Wide Pass. The pass costs 15,000 yen for adults (12 years and older), which is considerably less than the all- country pass JAPAN RAIL PASS). It is valid for three consecutive days, and thus, ideal for short getaways in the Kanto area.
My top recommendation for this pass is using it to get to Karuizawa (Nagano Prefecture), which takes just over one hour from Tokyo on the bullet train.
Karuizawa has great hiking, cool mossy forests to explore in the summer, skiing in the winter, beautiful parks with bright red maples in the autumn and (for shoppers) a huge outlet mall near the station. Book one night at a hotel or chalet, and you have a perfect two-day getaway in the mountains.
The pass also allows you to get to Nasushiobara (another mountain resort area, not far from Nikko) using the bullet train, as well as Utsunomiya (famous for its gyoza festival), Oyama (more hiking) and Jomokogen. Regular trains will get you as far as the Izu Peninsula (this takes some time), Mt.Fuji and its lovely nearby lakes, Narita (which has a famous temple, not just the airport) and all the way to Chiba (where Tokyo Disneyland is located).
While a little bit of research will be necessary, this pass really allows you to see all sorts of different sides of Japan.
For more detailed information, and special deals related to the pass, check out: https://www.jreast.co.jp/multi/en/pass/tokyowidepass.html
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