Vasa Museum - I think this is one of the best museums in Europe. Make sure to watch the film about how the raised the ship and take one of the Tours
Stadshuset (City Hall) - The guides always do excellent tours and you will get to see where the Nobel Prize dinner is held and learn about the city government
Skansen - If you are there at the beginning of August, visit on a Tuesday. On Tuesday evenings in summer, they hold Allsång på Skansen (singalong at Skansen - a show featuring many popular Swedish singers that is broadcast live) and during the day most of the exhibits are open and you can probably catch some of the dress rehearsal in the afternoon. If you are there later in August, go on a weekend.
The Abba Museum is a bit expensive but if you like Abba it is really fun.
Walk up to Fjällgatan on Södermalm for the great view.
I love visiting Östermalms Saluhallen. It is a beautiful, old-fashioned food hall with many traditional Swedish specialties.
When the weather is nice, my favorite place to eat is Rosendalsträdsgårdskafe
This is a wonderful little cafe set in the middle of a garden in an apple orchard. They have great organic salads, sandwiches, and wonderful breads and baked goods.
For day trips, I recommend a visit to Gripsholm Slott. You can travel by train switching in Läggesta to an old fashioned steam train. Alternatively, you can travel by boat from Stockholm. The castle is really interesting and the town of Mariefred is nice.
I recommend getting the 5 day Stockholm Card if you are there for a week as It is easy to get your money's worth. There are many good museums but the Nordiska and Fotografiska are among my favorites.