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. The airport has a good connection to the down with the Airport Express electric bus that takes you from the airport to the city centre shortly after the flight has arrived. It is also possible to go from the city centre to the airport with the same bus. Alternatively, ordering a taxi is also a possibility. For more information, see: https://www.tartu-airport.ee/en/transport/leaving-the-airport/
It is the main airport of Estonia (https://www.tallinn-airport.ee/en/). In 2018 it was named the best small airport in Europe. 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 should buy the ticket in advance on-line https://www.tpilet.ee/en/timetable/lennujaam/tartu Some buses would skip the station if no one has bought a ticket for the airport stop. However, even 15 minutes in advance is enough. The airport bus station has a self-service ticket dispenser where you can buy tickets for buses stopping at the airport on the way to Tartu as well. 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.