There are three possibilities to get to Tartu:
Directly to Tartu
There is a plane connection via Helsinki. Approximately 15 minutes after arriving of the plane a shuttle-bus is leaving from the airport to the city. The bus brings the passengers to their final destination. Further information: http://www.tartu-airport.ee/eng
By bus: From Tallinn airport leave several buses per hour directly to Tartu (ca. 2,5 h). You find bus schedules and tickets here: https://www.tpilet.ee/en (Please choose the destination "Tartu Coachstation") It is recommended to buy tickets in advanced as busses might be booked. (Tickets can be purchased also at a ticket-machine at the exit of the airport or directly from the bus driver, if empty seats are available.)
The exit of the airport is on the floor below the arrival hall. In front of the airport you find three stops: One for buses leaving Tallinn, one for buses to Tallinn and one for Tallinn city transport.
By train: It takes around 2 - 2,5 h to go by train to Tallinn depending whether it is an express train or not. The main station is located in the old-town another station "Ülemiste" is close to the airport (15-20 minutes to walk). However the latter might be difficult to find. Timetables you can find here: http://elron.ee/en/ Tickets can be easily purchased from the guard on board.
There are bus connections directly from Riga Airport to Tartu. You may also go by bus nr. 22 (timetable) to the main busstation, where you have a bigger choice of connections. Following bus-companies offer services to Tartu: Luxexpress and Ecolines.