Where to stay in Tartu? Few areas especially stand out.
Tartu bus station is located in very city center. Does your trip involve arriving by bus or departing by bus later? Then that's a valid reason to stay in the center or in the old town.
Old town in Tartu is about 5 minutes away from city center but to the other side of center than Karlova. Visiting the old town is a must when in Tartu.
This area of many historic wooden houses and interesting local bars might be the best part of Tartu to stay in, when you're more interested in exploring local bars and meeting locals than exploring the old town. Of course, Tartu old town is just a 15 minute walk away.
Near train station
Train station is a 20 minute fast walk or 30-40 minute slow walk away from city center. Google maps suggests 24 minutes. Staying in that area makes sense when you're arriving by train or leaving by train early in the morning.
Previous suggestions matter more when you're planning to get around by walking as I do. Tartu also has good public transportation system.
Taxis are a bit more expensive than in Tallinn. Still, a ride around city should not cost more than 5-6€. And traveling by your own car means that parking in city center and old town is for a fee.