Hejsan stratobee,
(jag är finsk så jag pratar lite svenska)
ESSB is doable as long as you don't need fuel. around 35EUR/landing, and around 30EUR/day for parking. If you need AVGAS, you'll be raped.
stratobee might disagree with me here, but I don't think there is much to see in southern Sweden when flying. There are many pretty places up north, but the GA, especially rental GA, possibilities are non-existent.
Stockholm is a very pretty city, but it is much better when explored on foot.
Hej din gamla finne!
I would agree, flying it's mainly a lot of forests and a lot of lakes down south. Stockholm best explored by foot. It's a gorgeous city. They call it the Venice of the North for a reason.
How long are you staying in Sweden/Stockholm?
Are you visiting any other places nearby or just staying in Stockholm?
Unsure how widely you have travelled around Europe, but if you haven't been to northern Germany, try to fit Tallinn (it's in Estonia, but it is very similar in style to many cities in northern Germany)to your schedule. Beautiful (very)old town, easily accessible from Stockholm by a boat or plane.
Avoid the drive from Gothenburg to Stockholm, it is the most boring 6 hours of your life.
Copenhagen to Gothenburg isn't anything special either, apart from the bridge to Malmö.

try to fit Tallinn (it's in Estonia, but it is very similar in style to many cities in northern Germany)to your schedule. Beautiful (very)old town, easily accessible from Stockholm by a boat or plane.
Tallinn is a very pretty city, and it is what an average American would consider "Europe". Stockholm is as pretty if not prettier, but not as "European" if this is your first European adventure.
Tallinn airport charges vary, no-one knows what they are, depends on the amount of nickel/dime you'll get.