The Asheville Mamas group is huge and very active and the best way to connect with hip mamas in Avl.
Lots of areas are family friendly, you'll have to be a little more specific :) What's important to you? Do you want high ranking schools? Do you want suburban kids playing in the street type neighborhood? Or do you want walkable to certain things? Or bikeable? Country/city? Or...? What part of town you and/or your partner work in would also be a factor.
We love West Asheville. I've lived in a very hip, walkable part (near bars, playground, stores, yoga, etc) which was really fun. Now I live still in West Avl, but a little further out, near Enka/Candler. We have an awesome huge yard now and still live close to everything (5-15 minutes to drive anywhere in town).
The Iwannas is an online and in print newspaper that is always how we find great rentals. Its a really cheap subscription to be able to see contact information for the listings. It would be another good place for you to start to get a feel for what house prices are typical for what areas.