M-shed is a free museum about the history and people of Bristol. My 5yo really enjoyed it - very interactive. It's on the dock, so near SS Great Britain etc., lots of food places and we even caught the little ferry from one side of the dock to the other.
M shed is great as is the SS Great Britain. If you walk around the harbour at the moment there is the 'Gromit unleashed' exhibition. Basically a trail of Gromit sculptures from Wallace and Gromit, all painted differently. We are local to Bristol but my kids went Gromit hunting last week and loved it! It's free!
Bungalow Inn is really nice and has a free softplay area and nice food, puxton park is lovely to for a day out so I have heard. Weston is ok, it's not the nicest of beaches but it's ok to mess about on have fish and chips and the pier is ok.