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.
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.