I recently went over and started at Gothenburg, went around the west coast to a youth hostel to the west of Uddevalla (on the water, great views, very relaxed, kids more than welcome, opportunity to hire kayaks and associated gear ... sadly, can't recall the name of the place! Will check and get back to you). From there we took a day trip to the fishing village of Lysekil, very beautiful, nice to wander around, and we had a fantastic lunch in a restaurant right on the docks ... on the way back to Gothenburg we stopped to check out the old fortress ruins at Kungälv, which has an interesting history, and is great to wander through (they also offer guides, we went without).

I know it's on the west coast, not the east, but depending on your plans you could either fly or take a train across from one coast to the other ...