A week to cover all three places isn't going to leave you much spare time!

Nagoya isn't that exciting, more of a regional industrial centre. Nagasaki of course has the Atomic Bomb Museum, which you should go and see: it's very sobering.

In Tokyo you'll probably have to choose between culture, food, temples, and gadgets. Gadgets are good...you need gadgets...gadgets are your friends... Also, it's a big place, and getting from A to B if A is one side and B is the other can take a looonnnng time. But assuming you are in the centre or thereabouts, then Akihabara for gadgets, Asakusa for culture/temples, Ueno for culture/museums, Shibuya and Shinjuku for food/beer/bars/general bizarreness. All these places are 30 minutes or so from Tokyo centre (= the imperial palace). Additional thought - if you are on a high level exchange program, and it's to a very Japanese environment, your hosts will expect to take you out every night for entertainment and beer. This will be a Good Thing in Nagoya, but might be a bit frustrating in Tokyo if you're trying to grab some time for yourself.

Regards

Mark