I made an overnight stop at Vasteras last year. Beautiful airfield, very friendly club and fuel service and minimal, if any, landing fee! (I actually can't remember, which has to be good news when visiting any airport!).
As said, the train ride is 1Hr to Stockholm and Vasteras is a very nice lakeside city to stay in.
I also visited Bromma, as said already very helpful people, a little more expensive (maybe 30 euro equiv. without overnight parking?) but right in the city.
One tip is to buy a landing season ticket at the first airport you visit - I recall about 50 Euro, but no further charges for a week or something. Would have been worthwhile, if I'd known about it! Don't know if it covers parking though.
When in Sweden a very scenic place to visit is Visby Island, off the coast some way S of Stockholm.