Japan has an extremely vast culture, and if you study abroad and stay in once city, then you will likely miss out on much of Japan's rich culture. While studying abroad in Tokyo, I personally made multiple trips to visit a couple more famous cities on the south side of Japan.
Tokyo is easily the busiest city in Japan, in that each disctrict could easily take up an entire section. Without going into too much detail, I'll list and describe some of Tokyo's more touristy districts in no particular order.
Shibuya is one of the most active districts, featuring Hachiko statue, Shibuya crossing, the 109 building, and more. Shibuya is often associated with fashion, but just walking down the street tells you that there's much more to Shibuya than just that.
Harajuku is a subdistrict within Shibuya with a strong focus on more niche fashion. Clothing stores tend to be more hip, more colorful, and in general highly geared to young adulty style.
Ginza is Tokyo's luxury district featuring high end department stores. In addition, Ginza is home to the Ginza Graphic Gallery, with free admission to the public.
Shinjiku is a commercial and business hub of Tokyo, featuring Shinjiku station, the busiest railway station in the world. Shinjiku features places like the Tokyo Metropolitan Government Building (which you enter and visit the observation decks for free) and in general a large amount of shopping centers.
Ikebukuro is another commercial and entertainment district. For female anime/manga fans, this district is very much your turf, as many of the anime shops around here are female oriented. Aside from that, Ikebukuro houses lots of shoping options incluing the department store Sunshine 60, which features places like the Pokemon Center and a very cool sky deck.
Akihabara, the anime capital, is undoubtedly my favorite place. Akihabara is surpisingly small at a couple streets, but stores are stacked and stacked upon each other, making each store incredibly dense in content and merchendice. Very notable are places like anime book stores, which can often sell things like artbooks, doujinshi, and of coures manga/light novels, character goods stores, Arcades, which are nowhere near exclusive to Akihabara, but are the best brances if you want to win something from the claw machine (more on this elsewhere), and second hand figure shops, of which you can buy quite nice figures for significantly more reasonable prices, of which would be hard to find outside of Akihabara, much less outside of Japan.
Yokohama is unfortunately the place that I spent the least time visiting. Located less than 30 minutes by transit south of the heart of Tokyo, often rivals Tokyo with the second largest populatiuon after Tokyo. Yokohama is famous for it's status as Japan's prominent port city. Listed below are some of the locations that I visited, as well as some that I did not have time to.
The Cup Noodle Museum is located only a short walk from Shin-Yokohama station. Not only does it describe the history of instant noodles, it also features a food court to sample noodles from around the world and an option to customize your own cup noodles.
Minato Mirai, directly translated to "future port", is Yokohama's upscasle skyline. Minato Mirai contains a lot of Yokohama's major attractions such as shopping malls and even a small amusement park.
Yamashia Park is a walkway along the sea that connects may of Yokohama's attractions.
Chinatown is another location that Yokohama is famous for. Considered Japan's largest chinatown, there is a wealth of shops and restraunts to experience.
Kyoto is undeniably the cultural capital of Japan. As a former capital of Japan, Kyoto has a wealth of temples, shrines, castles, and other historic sites. Below is a list of popular visitor locations within Kyoto in no particular order.