Djurgården is an island in the east of the city (technically a part of Östermalm). It can be reached by ferry or by tramline 7 "Waldemarsudde". You can take the tram from Östermalmstorg, at Nybroplan, where you can also see the Dramateatern building. Djurgården has some quiet areas as well as a lot of musea.
This is a place in a greenhouse where you can have lunch or fika and sit outside in summer. They serve food mostly from their own harvest and bakery. It's great.
They only have outside seating (I think), so it's more of a summer place. It is in a nice green area.
This is a museum which I've never entered but the gardens are nice and it is a scenic lookout point. It's a calm area that is really relaxing.
Some people like it, some don't. Personally, I think the Vasamuseet is very impressive. The Vasa is a ship that sank right on it's first voyage.