[GoNative]

I'd also recommend Hakodate for some great historical stuff. It's a beautiful city with one of the best night views of a city (from Hakodate yama) you'll see anywhere in the world. Goryokaku a magnificent star shaped fort which was the site of the last battle of the Boshin war is a must see if you're into history and going to Hokkaido.

As masaegu points out Otaru is nice port city with a good historical precinct and is beautiful in summer. Sapporo obviously has some historical stuff but like much of Hokkaido don't expect it to be the ancient history you find down on Honshu. Large scale Japanese settlement in Hokkaido didn't really occur until around the 1700's. Sapporo does have one of the biggest underground malls in the world (nice in winter when it's cold and snowy at street level).

With only a week I would stick to SW Hokkiado. Cities like Otaru, Sapporo and Hakodate and some scenic places like Shikotsuko and Toyako. These are two magnificent lakes which are amazingly beautiful in summer and both have little onsen towns to stay in. Noboribetsu is also pretty cool with some great volcanic features and is sort of situated between the two lakes.

I wouldn't try and see too much of the island in just a week as you'll be doing little else other than travelling. Hokkaido is pretty big and with no shinkansen it takes awhile to get anywhere.