Hey there,
I've been living and working in Japan for the past 9 months, and I've traveled extensively. I wish I could recommend just one area, but there are so many things to see! The most important thing about your trip is to narrow down exactly what kind of thing you're interested in. If you want shopping and city life, Tokyo and Osaka are where it's at. If you're looking for historical stuff, it's everywhere. However, to get the best for your time, my suggestion is definitely going to the Kansai region. This includes the cities of Kyoto, Osaka, Kobe, Nara, etc (but those are the big, famous four). Kyoto has a good balance of city with LOTS of traditional -- tons of temples and shrines to walk to, some of the most famous in Japan that even Japanese people travel to see. The plus to this is that cities like Osaka, Kobe, and Nara are only about 45 minutes to an hour by train in this area, which makes for really easy day trips or hopping around.
TRAVELING/MONEY: If you plan to travel by train (not the fastest, but the best), please invest in the JR PASS. You can buy this at a Japanese travel agency wherever you live. You pay a certain amount of money, and get access to however many trains you wish to ride for a designated amount of time. My sister paid $300 for one week when she came to visit...and just one ride on the shinkansen (bullet train) going one-way costs about $160 sometimes. Also, the Shinkansen goes all the way down to Kyuushuu, and it gives you access to all "JR trains," so if you plan to go to a lot of cities, it's your best bet.
ALSO: In Japan, almost everything is cash-based. In Tokyo, you'll have no problem using credit cards, but even in Kyotoa lot of people only take cash. It's really safe to travel with a lot of cash inside Japan, just be careful you don't lose it on the way in. I would recommend converting the money before you come, but that's up to you. By a lot, I mean that people usually walk around with like...$700 or $800 in their wallets and it's not that dangerous, so long as you don't leave your purse lying around (common sense).
If you get the chance, you should also check out Kyushuu, the southernmost large island. It's a bit warmer, and has tons of interesting stuff that people often over look (i.e. Kumamoto Castle, Nagasaki and Amakusa --the locations of a huge historical religious rebellion by Christians way back in the day -- Kagoshima)
These are my recommendations by city:
Kyoto:
Temples and Shrines:
Kiyomizu Temple
Ginkaku-ji
Kinkaku-ji
Yasaka Shrine
Fushimi-Inari shrine
Nijo Castle & the Imperial Palace
***Note: You need to make a reservation online for the palace, and you only get a 1hour guided tour of the perimeter; you can't go inside. Nijo Castle is much cooler, you can enter, and you can also purchase an English audio tour for about $5. No rush!
Kyoto Station: One of the most beautiful modern buildings in Kyoto, it draws a lot of attention from tourists in its own right!
GION CORNER: This is the name of an establishment that puts on a medley display of traditional Japanese arts, including tea ceremony, a geisha maiko dance, koto, bunraku puppet theater, etc. Great place to see a sampling, especially if you don't speak a lot of Japanese (a whole play can be mentally exhausting if you're not well-exposed to the language).
OSAKA:
Nanba, the Pokemon Center (if you're into that), Tennoji Temple, Sumiyoshi Shrine (reportedly the OLDEST Shinto Shrine in Japan, predating Chinese architectural influence...so ya know it's legit
KOBE:
Shopping and night-life; ALSO if you like Sake / Ume shuu, or are remotely interested, Kobe is famous for some of its old sake breweries. There are a few that will let you take free tours, and also sample some sake. "Kikomasamune" in particular has an English video that explains the age-old process they use in distilling, and has free sake samples!
NARA:
Daibutsukan: The Great Buddha Hall. This is a temple that allegedly holds the largest bronze Buddha in the world. Nara also has tons of wild dear roaming around that are people friendly, and which you can feed (you can buy the deer crackers for about $1.50).
There are many other temples and shrines. Nara is like a smaller, more homey Kyoto, and can easily be done in a day.
Kasuga Taisha Shrine (near the Buddha Hall) is a beautiful forested walk, and sells traditional omikuji (fortunes) in English.
TOKYO:
Akihabara: The electronics/manga capital of the world. If there's any electronic component you may ever want to buy, you'll find it here. Even if you don't want any, it's an impressive sight!
Harajuku and shibuya: Shopping and pleasant strolling! Also, Meiji Jingu Shrine is a beatiful place right by this stop that I recommend seeing.
Asakusa:
A great temple in this area, as well as the more artsy area of Tokyo.
Mt. Fuji:
If you want to go to Mt. Fuji, I have a few warnings. The best time to see Mt. Fuji is in the winter, it's said. A lot of times, the day may start out clear, but then it will get cloudy and the view will be obscured. If you're not climbing or hiking, you can see a view of it from the city Hakone, but it's kind of a tourist trap. If the clouds come down (like they did when I went), you won't see anything, and you'll have wasted a day (2 hrs there, 2 hrs back, plus waiting for the bus) to be in a town where otherwise there isn't much. Just make sure that you check the weather forecast before making any decisions regarding Fujisan.
PLACES IN KYUUSHUU:
K
umamoto Prefecture:
Kumamoto City:
Kumamoto Castle, one of the most famous castles in Japan with a stunning view of the city and beautiful grounds!
Mt. Aso: An active volcano that you can visit. Only a short 40 minutes from Kumamoto city. There are buses that go to Aso-san from the train and bus stations.
Eating Basashi: This is raw horse-meat...and some people are not okay with that, but it's a Kumamoto thing. You can, of course, find places with cooked horsemeat. If you don't want to eat it, don't worry, plenty of places DON'T serve it!
Kurokawa:
Hot spring town about 40 minutes away from Kumamoto city. You can get access to three hot springs for $12 (super cheap!)
Amakusa: Islands off the west coast of Kumamoto, which were the hiding place for many persecuted christians and the beginnings of the historical rebellion. You can actually see nagasaki to the north from the coast.
Nagasaki:
Lots of historical trade places, of course the peace park, and the man-made islands it's so famous for.
Kagoshima: Pretty far south in Kyushuu, this also boasts another live volcano, Sakurajima, which every day spouts TONS of volcanic ash. My car was parked 30 minutes away, for one day, and was doused with it. A pretty impressive sight!
PM is okay too 
