It is a small airport, served only by Finnair. You can book your flight directly to Tartu. It means that you change the plane in Helsinki. Tartu airport has a good connection to town with Airport Shuttle bus (http://www.tartaline.ee/airport-shuttle-eng), it takes 15 minutes to reach the centre of the city (bus takes you directly to the hotel, if you ask). Ticket costs 5 € (Euro is the local currency).
It is the main airport of Estonia (https://www.tallinn-airport.ee/en/). Also, it is served by some low cost airlines. You might like to visit the medieval old town of Tallinn (UNESCO world heritage).
Tartu is 185 km from Tallinn. There is a very good bus connection between the two cities, with at least two buses per every hour during daytime. The bus ride to Tartu takes two and half hours. Most of the buses have free wireless internet and other minor conveniences. The bus stops are located at the city side of the passenger terminal next to the tram stop. The ticket costs about 8€-15€ depending on the purchase time and departure time. You can buy a ticket in advance on-line https://www.tpilet.ee/en/timetable/lennujaam/tartu. The airport bus station also has a self-service ticket dispenser where you can buy tickets for buses stopping at the airport on the way to Tartu. The machine accepts card payments.
NB! If you are planning to arrive at Tallinn airport later than 23:00, please contact Airport Shuttle bus (http://www.tartaline.ee/airport-shuttle-eng), which takes you to Tartu, if pre-ordered (no later than 15:00 on previous day), for 20 €.
Another (less convenient, but maybe with some extra connections) option is Riga airport, 250 km from Tartu. The bus connection service is available four times a day, travel from Riga to Tartu takes 4.5 hours (https://ticket.luxexpress.eu/en/).
From St Petersburg you can travel to Tartu by bus (see: http://www.luxexpress.eu/en ). The bus trip takes 7 to 10 hours, depending on the bus you take.