Snow is only really a problem in the northeast and in Hokkaido. Most tourist areas you should be fine. Some of the best areas to see are Tokyo, Kansai (Kyoto/Nara/Osaka), and Hiroshima. Kyushu also is very nice.
You could spend a lot of time in Tokyo and still not see everything.
In Tokyo a few of the best places to see are the Tsukiji Fish Market, plus the site that every last tourist goes to see, the Sensoji Temple. If you are in Tokyo on a Sunday afternoon, be sure to go visit Harajuku to see the youth with their wild fashions. Meiji Shrine and Yoyogi Park are next to it and well worth seeing also. And be sure not to miss the Shibuya Hachiko intersection especially on a weekend when it looks like half the city is crossing the street all at once.
In Kyoto you could spend weeks there and not see everything. The three best sights though are the Kinkakuji Temple, Kiyomizudera Temple, and Fushimi Inari Shrine. Fushimi is the most time consuming; you could spend a couple of hours to over half a day if you want to romp over all the trails. And the Otagi Nenbutsuji is one of the most humorous and whimsical places you could visit in Japan - well worth visiting, even for kids.
Nara is often neglected by some tourists, which is a shame. Missing the Todaiji Great Buddha in Nara is like going to India and skipping the Taj Mahal. Nara Park has a lot of great places, such as Kasuga Shrine, Kofukuji Temple, and feeding the many deer in the area.
A few other great places missed out even by those who go to Nara though are the Isuien and Yoshikien Gardens. They are gorgeous and not crowded at all.
In addition, there are other numerous great places you can visit in the area. Okayama is not far and it is famous for its castle, plus Korakuen, one of Japan's Top 3, and also the Handayama Garden. If you'd like to get off the beaten trail for a while, the half day hike up to Konpirasan in Kotohira is stunning. Going to Takamatsu is also nice - it has one of Japan's finest gardens also, called Ritsurin.
Hiroshima also has a number of good places. Miyajima is the most famous, and aside from Hiroshima itself, Onomichi with its temple tour and Kosanji Temple is spectacular. Just beyond Miyajima also lies Iwakuni with its historical bridge and castle on top of the mountain. And if you really want something mind blowing, go see the Akiyoshido Limestone Cave in Yamaguchi - a 300 million year old marvel with an underground river going through. Hagi is also a delight.
For Fukuoka, there are many open parks, shopping, seaside historical areas, and urban neon. See this site as a good guide.
Don't miss the Nanzoin Temple which is one of Japan's finest yet almost totally unknown to the outside world and completely free.
You can also see a lot of previews on the best there is to see here.
In terms of history and varied points of interest, Nagasaki is really the most interesting city on the island. It is worth a couple days to see the city, and another day to see areas nearby like Shimabara/Unzen or whatever interests you. Gunkanjima, while it takes 3 hours, will blow your mind.