Oh my, what a great capital Stockholm is! Along with two friends, I visited briefly the most populous city in the Nordic countries and it was a jaw-dropping experience.
Stockholm is frequently referred as "The Venice of the North" and to be honest there are good reasons for that. The city itself is actually split between fourteen islands, the Old Town being built on Gamla Stan. Almost all of the islands are connected by bridges but the best way to visit the city is probably by boat. There are a lot of offers of boat tours but we prefered the "Under the Bridges of Stockholm" sightseeing tour.
Make sure you go on Gamla Stan island and lose yourself amid the most ancient buildings in the city. This is also where you'll find the Royal Palace and the Nobel Museum.
We were staying at Langholmen Hostel which has the particularity of offering, in the same building, a low-fare hostel, a nice hotel and a prison museum. The place is just crazy as you are sleeping in actual cells that have been restored. There are a lot of pictures, objects and explanations about the history of this building. The staff was welcoming and there is a all-you-can-eat breakfast.
Go there and explore, preferably in summer time! During our stay the weather was nice, allowing for pictures with crazy, cloudy skies.