I went to Stockholm this year. If you're doing touristy stuff, I'd recommend getting the Stockholm Card which gives free admission to most museums and attractions. It also includes free public transport.
I went in August, so my experience is very different. I really enjoyed Skansen, which is a big open air museum with a very good zoo. Obviously, it's open air, so would be better in Summer, when they also have a weekly pop concert.
I found the main royal palace a bit unengaging. All the little signs as you went round told you about the things in each room, as opposed to what happened there and the stories of the people who used them. I found the royal armoury museum tucked round the side much better, because it taught me the history of Sweden. I might have enjoyed the palace more if I went to the armoury and the other two museums first.
I also took a boat trip to Drottningholm Palace, which is incredibly beautiful. They have regular guided tours of the palace itself, which are a must if you go there. Unfortunately, this was my last day and by then I was shattered and didn't see much of the other bits.