The parkland at the Bishop's Palace is nice on a sunny day for a walk and picnic.
Durham University Botanic garden is very child friendly
Locomotion railway museum at Shildon (just outside Bishop) is great and free to get in
good soft play at darlington www.adventure-point.co.uk
a bit further but Preston Park museum in Stockton is fab; my two wanted to stay all day
Hardwick Park at sedgefield is another good (free) day out www.durham.gov.uk/hardwickpark
Thorpe Farm centre at Greta Bride, Barnard castle www.thorpefarm.co.uk
Raby Castle www.rabycastle.com
Bishop town hall often has special events for children (i.e plays, crafts puppets etc) www.bishopaucklandtownhall.org.uk
Hamsterley Forest is great to wander round and explorer
Fulling mill and oriental museums have drop in activities for children www.dur.acuk/fulling.mill
Durham is a very beautiful city with cathedral (features in Harry potter films) and castle but bear in mind the streets tend to be cobbled, narrow in parts (and steep). Up towards the catherdral are a a few nice, interesting shops (Mugwump, Georgian windows) where you can have a rummage about for gorgeous stuff (not particulary child friendly due to size and stock but nice for you) Lovely walk along the riverside and you can admire the hulking students boating on the river!