The main air gate to Estonia is Tallinn Airport, where a direct bus leaves to Tartu every hour. The travel takes about 2.5 hours.
Tartu has regular bus connection network to Tallinn, Riga and Saint-Petersburg. Information on schedules, prices, is available online at T-pilet, where also tickets can be purchased. The bus station is located in the centre of Tartu near the Dorpat conference centre.
As an alternative, there are also trains travelling between Tallinn and Tartu several times per day. You can find time tables and ticket information on http://elron.ee/en/. The nearest station from Tallinn airport is Ülemiste (ca. 15 min. walk).