There are a lot of ways to get into Tartu. You can come by car or by bicycle, or use public transport.
There are frequent buses from Tallinn to Tartu, departing either at the bus station (timetable) or at the airport (timetable). Bus tickets can be also bought online. The journey takes around 2.5 hours.
There are also 4-6 trains/day, express train takes 2:15, slower 3 hours. Online tickets and schedule: Edelaraudtee.
There are 4 daily buses between Riga and Tartu that depart Riga at 8:40, 18:45, 22:45, 23:55. The journey takes around 4 hours. Tickets can be bought at luxexpress.eu. Riga is called Riia in Estonian language.
By train: 13:10 from Riga central station to Valga, connecting train 16:42 to Tartu (arr. 18:12). Summer schedule may change.
By ferry via Tallinn, or by direct plane to Tartu airport.
By train via Pskov: The night train to Pskov departs Leningradsky Railway Station at 18:30 and arrives to Pskov at 7:30. Walk 200m from the Pskov Railway Station to the Pskov Bus Terminal. Take the bus to Tartu that departs at 8:20 and arrives to Tartu at 10:50. There is one more bus from Pskov that departs at 14:00 and arrives to Tartu at 17:20. A ticket for this bus can be bought online. Pskov is called Pihkva in Estonian language.
By train via Narva (not recommended: quite expensive; passengers are waked up at 4am for the border control). The night train to Tallinn departs Leningradsky Railway Station at 18:05. Get off in Tapa at 7:19. Take the train to Tartu, that departs Tapa at 7:48 and arrives to Tartu at 9:29
By train via Riga: 19:00 from Rijskaja, arrives 09:45 Riga. See from Riga section.
By plane via Tallinn: 3 airlines offer daily flights between Moscow and Tallinn: Aeroflot, UTair and Estonian Air.
By bus: There are 4 daily buses between Sankt Peterburg and Tartu that depart at 00:15, 11:55, 20:00, 21:40. The journey takes around 7 hours. The tickets can be bought at luxexpress.eu.
By train: there is no convinient connection to Tartu, you will have to wail till next morning in Narva, Rakvere or Tallinn and take a bus to Tartu from there. The daily train to Tallinn leaves Vitebsky Railway Station at 17:22.
via Tallinn (2.5h by bus/train) - Ryanair (London, Germany, Dublin, Milan, Oslo etc), Estonian Air and others.
via Riga (3.5h by bus) - AirBaltic, Ryanair (many destinations) and others. Bus directly from Riga Airport to Tartu departs 17:45 and 22:00, other connections from Riga see above.
via Helsinki - 1.5hr boat to Tallinn (10+/day) or Flybe direct flight to Tartu (1/day)
via Stockholm - overnight Tallink ferry to Tallinn departures 18:00
by train - via Moscow, at least 2 nights adventure. Visa needed. See From Moscow section.
by bus - Ecolines or Eurolines, both via Riga