If you are a public transport nerd you should definitely try the brand new lightrail tram, it takes about an hour from Keilaniemi to Itäkeskus (or the other way) and shows you some less central parts of town.
As a librarian, I must of course also say that our libraries are cool. The one I work in is the first library to be planned and built for public use, in service since 1882 www.helmet.fi/en-US/Librar...
If you are into art, then the Diedrichsen museum is a personal favourite based on a private collection, located in a very nice part of town www.didrichsenmuseum.fi?lang=en