I have just arrived at Stockholm and I'm eager to see galleries and exhibitions of Swedish design.
Apart from the National Gallery, what are good places to start with?
I'm interested in graphical design, typography and industrial design as well.
Ok - seems like I need to answer anyway. The answers above are on the main tourist track and I don't think that was the question really. There is no museum in stockholm dedicated fully to swedish design although several touch on the topic more or less all the time. (In Gothenburg there is Röhsska museet that is more on topic.) Nordiska museet - for example - had a wonderful exhibition about plastic design last year and are now showcasing silver jewelry by Rosa Taikon. It is an enormous building located when you enter the island Djurgården that was built in the national romantic style of a hundred years ago.
Tekniska museet sometimes has a main exhibition with vehicles from a time gone by. The first versions of Saab are up for display. Sometimes it also got temporary exhibitions that touch on industrial design.
Photography has recently gotten its own museum called Fotografiska in an old toll house in the harbor of Södermalm. The building was first meant to be used as an abba museum.
Moderna museet is mainly a modern art museum but sometimes got exhibitions bordering to the design theme (defining the border can be hard sometimes - thinking about the andy warhol exhibition some years ago).
A gallery I haven't visited yet but heard a lot good about is Färgfabriken. It should be worth a visit just because it is located in an old factory building. But they also have interesting exhibitions - now there is one about norwegian form.
If interested in how to create a house to be used for art there are two really nice smaller art museums at Djurgården that has partly been built by private persons about a hundred years ago to incorporate their art collections. The more famous one is Prince Eugens Waldemarsudde where you can occasionally see very modern art and photography but typically has paintings from around 1900. Now they are showing Carl Fredrik Hill - a personal favorite. The other one is called Thielska Galleriet and is located at the "end" of Djurgården. They have a very nice collection of what is called "sekelskifteskonst" here. I do not know if there is a word for it in english. Art from around 1900.
I make this into a community wiki so that smaller places can be added over time.
Right, first up, the Vasamuseet (Vasa Museum). It displays the only almost fully intact 17th century ship that has ever been salvaged, the 64-gun warship Vasa that sank on her maiden voyage in 1628. It didn't even make it out of the harbour ;)
This is one of the most amazing museums I've ever been to, and it's an absolute MUST NOT MISS when you visit Stockholm.
In the Southeast is the Vaxholm Castle - a coastal defense museum.
On Lidingö island is the Millesgården sculpture museum - see below:
There are a bunch of museums all around Stockholm (did I mention the Nobel museum? - another must-see sight of Stockholm). However the costs could quickly add up, so I'd suggest getting a Stockholm Card. The Stockholm Card allows free public transport as well as free admission to 80 museums and sights in Stockholm, free sightseeing by boat and other bonus offers. Adult 24 hours 425SEK, 48 hours 550SEK, 72 hours 650SEK, 120 hours 895SEK. Children (7-17 years of age) 24 hours 195SEK, 48 hours 225SEK, 72 hours 245SEK, 120 hours 285SEK.
Skansen is a really great outdoor park/museum, and you get an idea how the lifestyle and industrial design changed through the last 500 years, if that is something you're looking for.
Don't miss the Vasa Museum, even it's not exactly in your area of interest - you wont see an old ship like this in such a good condition and detail anywhere else.
Here is a good list of Stockholm Museums. I have only been to a few of them in walking distance from city center, and they where generally quite good. The aquarium was a bit disappointing imho. Not bad but quite small, and i would have expected more for the money.
Don't forget the world's biggest modern art museum: Stockholm metro.
On the wall of a station (I don't remember which one), I found a music score of La Marseillaise, the french national anthem. Maybe an homage to Bernadotte.
One building that is a monument to Swedish design that is technically not a museum is Stockholm's new City Hall (Stadhuset).
But it is a "museum quality" exhibit. It is constructed of brick, but laid out using "old-fashioned" (for the early twentieth century) artisan techniques. It was inaugurated June 23, 1923, the 400th anniversary of King Gustav Vasa's capture of Stockholm
Besides housing the offices of the city's administrators, important features are the so-called Blue Room (originally designed to be in blue), where the Nobel Prizes are awarded, and the Gold Room above, inlaid in gold, where dancing takes place after the awards. Also, there is a tower, topped by three crowns, the symbol of medieval Sweden. From the top, one gets a great panoramic view of Stockholm.