Part of Stockholm's extensive transit system is the commuter train network connecting central Stockholm to the other municipalities and suburbs surrounding it.
Depending on how far you are traveling, the commuter train is a little more expensive than regular subway fare, but it is still very reasonable.
Many of the day trips we have pointed out on this website can be reached via commuter train, and it is also connected to the airport. (Albeit slower than the Arlanda Express train). If you have the time, take a look at the map, and explore the suburbs of Stockholm!