One of my first recommendations concerning how to spend your time in Stockholm is to go watch a Fotboll game by one of the three football clubs located in Stockholm. Yes, there are actually three different clubs all competing in the same league:
AIK Solna (http://aik.se/start/)
Hammarby IF (http://www.hammarbyfotboll.se/)
Djurgården IF (http://dif.se/)
A week ago I went to my first swedish football game, Djurgården IF vs. Malmö FF and had a real great time. If you are an European you probably would argue that the Swedish league is not a well-known league in Europe at all, neither is there are big, successful Swedish club besides maybe Malmö FF, but beside this it is still worth it to go and watch a game here in Stockholm.
And therefore let me tell you something: Yes, probably the Swedish football league is not one of the strongest ones in Europe, but the Swedes themselves can get really patient about their teams and the games. And this is also one of the reasons why you should go there: To see how the people are dedicated to supporting their club unimportant to the actual result and the overall skill. These 90 minutes are one of the rare moments a Swede can get really passionate and emotional. You can probably have a really hard time trying to experience something equally in the matter of intensity and emotion.
But beside this phenomenon, it is also a chance to feed your Stockholm knowledge about architecture with help of the impressive two newly build stadiums, which are the Tele2 arena (opened in 2013) with 33,000 people capacity or the Friends Arena (opened in 2012) and at the same time the National Stadium with a capacity with more than 50,000 people.
Hopefully my short overview, pictures and explanations convinced you to also give this experience a try and join one club for a game. But you should better hurry, the season is rapidly coming to an end and also: the colder it will be outside, the colder it will be in the stadium ;).