I agree with everything LeMesmer suggested and must say that Skansen (you can see wolves AND wolverines!) is fantastic. We have a yearly pass as I have family in Stockholm. DD loves it!
Boat trip is essential, the archipelago is gorgeous in the summer. Fjäderholmarna are only a short boat ride out and have lovely little cafes, craft shops, a play area for kids and gives you a taste of the archipelago without having to do a long boat trip.
A good tip is to eat your main meal in the day, a lot of places do a special 'dagens lunch' (daily lunch) which includes a decent cooked meal, non alcoholic drink, salad, bread and coffee. usually around 90-100 krona. 'Max' do the best ever burgers is you fancy a Swedish burger, reasonably priced and made for you as you order it (pricewise it is on par with McDonalds). There are also lots of 'Grills or Sibylla Grills which are little huts that sell korv (Swedish sausage) at reasonable prices. If you aren't on a tight budget a 'Toast Skagen' (prawns in dill and mayo) is one of my favourites, as is a 'plankstek' . can you tell that I am missing Swedish food?!
Any alcoholic drink should be purchased from Systembolaget (who have monopoly on selling alcohol) and perhaps consumed with a picnic or in your hotel room as it's very expensive to drink out... Some places do a 'Stor Stark' deal on pints of lager, if you like a cheeky drink.
If you like a beach there are some in the centre although you may have to ask for directions. Smedsuddsbadet is good, lovely sandy bit by water and big grassy area to sit on. Child friendly and about 10 minute walk from underground stop Fridhemsplan, a pleasant walk through RÃ¥lambshovsparken will get you there.
Need to see to DD now but will post if I think of something else!