Stockholm has the only postal museum in Sweden, located in Gamla Stan (Old Town). This one has been opened almost one hundred years, so Sweden has a lot of postal history. Further, the old building the museum is in has been owned by the postal service since 1720.
The museum has two main parts. One part focuses on historical stamps and letters, and the other part has exhibits on how mail has been delivered over hundreds of years.
This is not a big museum, and there is no entry fee. I saw everything in less than an hour. You could stay longer if you have a very strong interest in the postal service, and its history. Location is about 2-3 blocks from the Gamla Stan train station.