The taxi stop is located in front of the terminal. The taxi might cost around 10 EUR to the city center and you can ask the airport staff to order one for you.
From Tallinn Airport to Tartu: Every hour, there is a bus which also take passengers from the Tallinn Airport (stop "Tallinna lennujaam" in Estonian) on their way to Tartu.
Trains for Tartu from Tallinn train station leave 3-4 times a day, and it takes 2 hours to get to Tartu. The train station is situated just outside the Tallinn Old Town and sea port.