By plane
Tiny Savonlinna Airport IATA: SVL is served by Flybe (http://www.flybe.fi/), which offers flights from Helsinki to Savonlinna up to two times per week day, and daily during the Opera Festival. Advance fares start from €32/one way, cheaper than the train, but go up quickly as the plane fills. There is a shuttle service to and from the bus station, which departs one hour before the flight leaves.
By train
There are 3-4 trains daily from Helsinki or Joensuu to Savonlinna. The price is reasonable especially for students, seniors and juniors. It takes about 4 and half hour to Helsinki by train. (http://www.vr.fi)
The main station in Savonlinna is called Savonlinna-Kauppatori next to the market square. The "station" is unmanned, but you can buy tickets in the train with no surcharge if you board here.