The countryside around Savonlinna has quiet country lanes and gently sloping hills, and is terrific for bicycle touring. Bikes can be carried on-board lakeboats for a small fee.
See http://visitsavonlinna.fi/en for information about canoe and kayak hire in and around the city.
Numerous operators run tours allowing you to explore some of this Lakeland area – head to the Satamapuistonkatu (passenger harbour) to see what's on offer. Most offer one-hour cruises around town between June and August (adult/child €15/7). You can also board one of the historic steamships operated by VIP Cruise for a 90-minute shoreline cruise (adult/child €20/10).