In Sunderland Barnes/High Barnes area is a nice family area and there are quite a few primary schools in that area (Plains Farm, The Broadway, St Mary's and Barnes that I can think of.
Barnes School is good, a few people I knows kids' go and they are happy with it. I'm pretty sure it has a web site which had the after schhol clubs on when I was deciding on schools, www.barnesjunior.co.uk.
Also Ashbrooke/Queen Alexandra Road area is nice but more expensive, good schools would be Hill View or St Mary's and there a few private schools. Close to Barnes area, nice park and 5 mins to city centre.
Over the water (so very near the beach and the St Peters Campus part of Sunderland Uni) Roker/Fulwell areas are nice and Fulwell School is supposed to be good, Whitburn is a bit further along the coast but also nice.
Parts of Doxford Park, Moorside, Silksworth and the Herrington areas are all ok but a bit furher from the centre and I don't know which are the good schools in these areas.
The areas I would avoid are Hendon, Pallion and Pennywell Sunderland South and Hylton Castle, Witherwack, Southwick Sunderland North.
Newcastle I only know that Jesmond and Gosforth are nice but expensive and no idea about schools. Durham is lovely and no further or worse for travelling into Sunderland than Newcastle. Personally if you are going to go to Sunderland Uni I would definitely look to rent in Sunderland (but go shopping through Newcastle!).
Hope any of this is some help.