Probably the most well known and popular national park of Finland among kayakers is Kolovesi national park. About one third of visitors arrive at the park by kayak/canoe or boat, not by cars! If you are an experienced paddler one good place to start is Oravi village by lake Haukivesi (part of Saimaa) from where you can paddle either through Linnansaari national park along lake Haukivesi or shorter way through lake Joutenvesi (lots of protected islands). Either way is beautiful. There used to be a carry-over path from Joutenvesi (through lakes Ruunavesi and Vääräjärvi) to lake Käkövesi but last time I visited the route was blocked by private lot so you might have to use the northern access. From Oravi you can rent a kayak and shop last supplies.
If arriving with a car there are two points, Käkövesi beach (a few kilometers from the park) and Kirkkoranta beach (in the park), see the map.
No motor boats (except few locals) are allowed at the park which offers rare experience of silence in the popular lake area of Finland.
For camping there are several sites but quite few good spots per site so if you move with a bigger group it is not guaranteed that there’s room or even spots for everyone. I guess Syväniemi site is suitable, I have seen one big (loud) group staying there when passing by. Camping in the park is allowed only at these sites.
Be aware when paddling especially early in the mornings and late in the evenings, the park is inhabited by Saimaa ringed seal, see pdf, the most endangerad seal of the world. You might get a glimpse of it in the water or even lying on the rock by the shore.
Kolovesi has very rocky shores, not too many sandy beaches there. The lake area is quite narrow so it’s suitable also for canoeing. Here are some pictures of my trips to Kolovesi from previous years.