I live here in Durham and know it well, and this isn't an easy question as some necessary information is missing most importantly your price range as depending on this you may not be able to get exactly what you're looking for. Also how old are your kids? There are a number of good elementary schools in Durham, and some good high schools, judging only by grades, none of the middle schools are stellar.
If you're interested in good hard data on the schools I highly recommend http://www.NCReportCard.org.
In terms of neighborhoods that offer good walkability there are several in Southwest Durham, there's Trinity Park area, (in between Duke and Downtown) there's the areas around Ninth St.
Having lived many places, and having spent the last 8.5 years here in Durham, I'd have to say Durham isn't the most walkable city. That said I personally love Durham. I live in the Duke Forest area by Duke. Large lots, and I'm within a 7 minute drive from Downtown, Southsquare, Brightleaf, Ninth St etc.
I'd be happy to speak with you and answer any questions you might have about Durham as well as the different price points in various Durham neighborhoods. Fell free to contact me through my Trulia profile. I've also attached a link to my website and Blog where there a wealth of information about Durham and the local area.