You're coming in June? Then you should not miss the late evening photo opportunities. In Stockholm, it does not get truly dark until 23.30 in June, so use the evenings! If you get off at 17.00 hours, you'll have at least 5 hours of good light left.Great post, I didn't think of all the places as it's almost three years since I lived there. Yes Fjällgatan is very beautiful with great views.
If you are into architecture, Stockholm is a virtual goldmine, with buildings ranging from 1100 to modern times. Old town is ofc great for the older buildings (another place is Östermalm/Vasastan), but everywhere else it is a mix between old and new.
Vantage points include Fjällgatan (which also have a nice cafe open until 23.00) on Södermalm, Västerbron (between Hornstull and Kungsholmen) and both Södermälarstrand and Norrmälarstrand (close to the water).
Everywhere you go in Stockholm, the water is not far away, so take it into account. A good way to go around is the sightseeing boats (so you can get pics from the sea).
Stockholm is also FULL of churches, so don't miss them! They might be hidden away, like on Södermalm, or very visible, like Riddarholmskyrkan. They are also almost always open.
The Royal Castle, with the changing of the guards is a given photo op, and if you want some history, you can go into the Livrustkammaren Museum to see some cool medieval clothes and armory (beware, it is dark in there).
You can also easily get to the Riksdag (house of Parliament/Congress/Government) which sits right next to the Castle, on it's own little island.
If you arrive around the 7-10/6, you might also catch the "finish school" carnival, as it is tradition for 18-year-olds to gather on the back of a lorry/truck and honk, play loud music and get drunk for the afternoon. This is typical for the second week of June.
Well, that is some suggestions. I would also recommend Djurgården, with the living museum Skansen, and the Vasa museum.
So, bring a UWA+tripod and/or some fast glass for indoor shooting in churches/museums and you are set. Welcome to Stockholm!