I live in Stockholm, so hopefully can help.
Depending on where you are staying, you will probably want to buy the 72 hour transport ticket, this will get you on all SL public transport which includes two boats in the city. I suggest at a minimum taking the boat from Djurgården (island where Vasa museum is located) to Slussen. You get great views of the city from that boat.
If you want to go out to the islands you want Waxholmsbolaget boats. They go from the centre of town and you can buy the ticket on the boat. The timetables etc are in Swedish, but just use Google translate to understand the website. You might find that this time of year the outer islands are pretty closed up as it's not the season yet.
You might be better off just taking the boat to Vaxholm which connected to the mainland and a slightly larger town. The trip up is gorgeous, and if you have bought the SL transport ticket and want to save money you can get the bus back into Stockholm instead of paying for a return ticket on the boat. waxholmsbolaget.se/visitor-information.
Alternatively there's a very attractive commuter boat trip which goes inland on the lake to Ekerö (Line 89, leaves from just outside the City Hall). That is free with the SL transport ticket. The trip is nice but the destination is just a standard Swedish small town. Timetable is here: sl.se/ficktid/vinter/H89b.pdf
Several museums are free for under 18s, and some are free for everyone. The Medeltids museum (middle ages) is underground next to the palace, free and pretty interesting. Vasa is definitely worth going to as you will only pay for adults.
If your kids are into video games the Tekniska museum is also pretty cool, with a massive games exhibit. Not sure which days you will be in Stockholm, but it is free entry on Wednesday evenings. Otherwise it's 120 Kr per person so a little pricey.
Skansen is nice if you would like to see the Swedish animals - bears, wolves, reindeer etc. But as chemenger said there's not always so much to see with the old houses. Possibly only worth going if you are interested in Swedish history or architecture.
I think there's plenty to keep you busy in the city for three days. If you are set on a walking tour there are a few "free" ones. They don't charge you to join, but you are expected to tip the guide. I don't know how much is sensible to tip, but I have a friend who has run a few tours and they seem quite interesting.
Eating out is very expensive in the evening. Weekday lunch is usually a good price for a few set meal choices in almost every restaurant. I would also suggest "Fika" which is coffee and cake, and a Swedish institution. Lots of bakeries around to try cinnamon buns etc.
Hope that's helpful.