I was in Stockholm for a week just two weeks ago visiting my girlfriend who was at home for summer, it's such a lovely city!
I recommend going to the Vampyr Lounge for some epic cocktails although Monday is the best day for this when it's like almost half price ....
Alcohol is very expensive in Sweden, especially out at bars so prepare for that!
I've been to Stockholm a couple of times and would recommend:
-Bus tour with audio, good to get to know the area
-Old town "Gamla Stan" for a bit of eating and drinking (expensive mind)
-Vasa Museum
-Strolling around by the water
-Skansen
Sthlm is a bit of a weird place in that while I quite like it, there isn't really anything outstanding that I can put my finger on and say YOU MUST DO THIS!
My girlfriend works at Skansen over summer
Although in the Aquarium part where all the animals are!
Agree on Gamla Stan, that's a must ... also, walk around some of the nice shopping streets: Kungsgatan down to Stureplan and then down Biblioteksgatan... pretty nice!
Then there is
Strandvägen (the most expensive property street in Sweden) which takes you towards
Djurgården (where Skansen and Vasa Museum is...)
The Stockholm Opera House is amazing too if you are into that (I am!)
Drottninggatan is another shopping street and there are loads of really great restaurants in Stockholm (some around Stureplan) ... just worth wandering around finding places really!
Transport wise, you can buy day or weeks pass cards that let you use any busses/trains/underground or trams in the city (you can get them at "post offices" or at stations... kinda like London oyster cards but you don't put money on them. I got a week pass for around £20!!