In Hiroshima, visiting the A-bomb memorial is a no-brainer
http://en.wikipedia.org/wiki/A-Bomb_Dome. That's why most people go to Hiroshima, so I'm sure that will be your first stop. Also, take a couple hours to go through the nearby memorial museum to check out some truly horrifying history. (And ride the old-fashioned trams. Two are still in service from before WWII) But you would be doing yourself a major disservice if you didn't take a day to visit the nearby
http://en.wikipedia.org/wiki/Miyajima,_Hiroshima Miyajima Island. You can reach it by a train ride and then a ferry. On this island is a small, old-fashioned seaside town lined with charming wooden shops selling grilled eel and maple-shaped cookies, which they are nationally famous for. There's also free-roaming, domesticated deer that are awesome to get close to and pet at first, but if you're too friendly with them they'll start eating anything you hold in your hands... camera, passport, your lunch, you name it. On this island you'll see some great temples, pagodas and you'll see Itsukushima Shrine
http://en.wikipedia.org/wiki/Itsukushima_Shrine , especially the famous orange torii gate that appears to float above the water. It is regarded as one of the three great sights in Japan.
Tokyo is a glittering metropolis, full of neon and skyscrapers. Shinjuku is like Tokyo's version of Times Square, with building high television screens, videogame arcades, pachinko (don't ask) palaces, karaoke bars and sushi-trains
http://en.wikipedia.org/wiki/Shinjuku. It's fun, but I'm more for the historical sights. If you love sushi you have to go to the largest fish-market in the world .... Tsukiji fish market
http://en.wikipedia.org/wiki/Tsukiji_fish_market Get there early enough and you'll get tuna fat straight off the bone. There's tons of restaurants nearby, but if you want to try the best there, prepare to wait in line a while (it's worth it though). Don't worry about those crap rumors about radiation. The fish is 100% safe.
For sights I went to the Yasakuni Shrine
http://en.wikipedia.org/wiki/Yasukuni_shrine which is a complex with an epic shinto temple, a museum and statues of kamikaze bombers. It's controversial because within it are entombed some class-A war criminals. But you're not a visiting head-of-state, so there shouldn't be a problem. I also went to the Sensoji temple, which is the most popular temple in the Tokyo area, though I found it a bit overrated and crowded. Still worth checking out.
http://en.wikipedia.org/wiki/Sensoji_temple. One of the best nights I had there was seeing a kabuki show in Gion
http://en.wikipedia.org/wiki/Kabuki-za though unfortunately I think they're undergoing some building renovations right now. If you're into your electronics and manga visit Akihibara and see beautiful girls dressed up as anime characters and electronic gadgets that won't be released in the US until 2015
http://en.wikipedia.org/wiki/Akihibara . If you're looking for some debauched nightclubs and ex-pat bars, head to Roppongi Hills at night
http://en.wikipedia.org/wiki/Roppongi_Hills but honestly I found the area seedy. Also if you get restless, go for a run around the Tokyo Imperial Palace
http://en.wikipedia.org/wiki/Tokyo_Imperial_Palace . There are tons of temples and sights to visit that I haven't listed, so do some research online and get a Lonely Planet book to help out.
Dude, I'm tired from typing all that. So chew on all that info for a while, and I'll tell you about Kyoto and Osaka next week. Hope this helps!
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