You can reach Tartu by flying to the local Tartu Airport which is directly accessible from Helsinki. Helsinki-Tartu flights are operated by Finnair seven days a week.
You can also reach Tartu through Tallinn and Riga. Tallinn International Airport is located just outside the city centre, on the road to Tartu. From Tallinn Airport you can take a Lux Express bus to Tartu (departs from the city center and also from the airport at least once in every hour, the journey time is 2,5 hours). It would be better to buy a ticket in advance from the previous web page. You may also take an Elron train from Tallinn to Tartu (departs also from near the airport, journey time for the express trains is 2 hours). Also, buying tickets in advance from the web page is highly recommended.