In fact, Tartu is the oldest city in Estonia and dates back to 1030. Sadly large parts of the town were destroyed in WWII, but Tartu still has a charming old town surrounded by large parks (some of which used to be fully built-up neighbourhoods before the war).
If you want to do nothing while in Tartu then that is just fine and sitting around the Town Hall Square and having coffee is a wonderful experience here. Since you are right there I do recommend you check out the kissing couple fountain. I went ahead down the road to see the 14th century St John’s Church, whose most outstanding feature is its 200 (from over 1000) terracotta figurines surrounding the church’s exterior.
Another place I saw was the University of Tartu, which is the national university for Estonia and was established in 1632 (although there have been periods of inactivity). All around the campus of 14,000 students you will find historic building, bridges, and other items. In fact, one of the reasons I traveled to Tartu was to see one particular building on the university campus, and that was the Old Observatory. But I am going to wait until tomorrow to tell you why. 🙂