If it's nice weather at the weekend, simply mooching about the harbourside is good fun (DS likes this, just coming up to 2). There are ferry trips round the harbour, replica of the Matthew (ship Cabot sailed to US in 1490summat) also the Great Britain. Lots of nice cafes etc.
Also Millenium Sq, which is by the @Bristol site has fountains and lots of running about like a loon space.
The museum at the top of Park Street is fab and free. Very family friendly with a small activity area by the cafe that will keep them busy and entertained. There is a great natural history section and lots of little tables with colouring sheets etc. dotted around. Mine love it, and never get bored of going there.