Tartu is an old university and Hanseatic city.
Though most of it's older buildings have been destroyed in wars, there is still a lot to see.
There are two city regions of old(looking) wooden houses : Karlova and Supilinn. If you haven't walked around there, you haven't been in Tartu :)
There's the town hall and the square with a fountain called "Kissing students". Here is a webcam focused on the square and as a bonus you get the weather predictions.
There are many museums : National Museum, Toy museum, Zoology museum, Geology museum (both of the latter are situated in Vanemuise 46), Botanical garden etc.
Often something happens in Tartu. Most of that information will be displayed on Tartu's homepage.