I agree with alot of what has been said here. If your move is only a temporary situation, and you have the resources, then I recommend an International School. However if you intend on fully integrating into your community (swedish I mean) then a local school would be the way I would go.
Out move to Sweden, is intended to be a permanent one, or for the length of the childrens schooling anyway. The boys were 5 and 2 when we first arrived, not too disimilar to yours. They both attend local swedish schools and picked up the language within a matter of months. They both speak fluently and unless told, people aren't even aware they are english until they speak to us! They have never had any problems with learning the language.
As for you, my advice would be to learn swedish asap if you intend to stay. It isn't a particularly difficult one to learn, despite what you might think if you watch Wallander. The difficulty arises when people want to speak english to you, which is often. The level of english spoken here is extremely high, so in one respect it is easy to get by, but you must be persistant if you want to become fluent.
Most Swedish women work, so life can be difficult for a sahm. I have a mix of friends, both swedish and expats, wohm and sahm, but it does take time to find the locals. One website I have found invaluable for meeting people is mumsinsweden.com.
Sweden is no Utopia and comes with its own problems but we love it here and have no intention of returning to the UK any time soon. Unfortunately I have no experience of Gothernurg, only the eastern coast, but please feel free to ask anything else.