hello! i have lived in wilmington for 3 years, so PM me or repost with any specific questions...i am happy to help.
what i can tell you, since you have mentioned this twice, is that wilmington is not a very pedestrian friendly city... its very much a car culture, unless you live in the heart of downtown. we actually moved to the center of town for this very reason, so i could walk to get groceries or a bite to eat, and although i am technically in walking distance, the lack of sidewalks and crosswalks makes it very difficult. There is alot of road congestion due to lots of cars, lots of people moving here, and roads that were made initially for a small town. They expand them, but it takes forever.
they did just put a beautiful bike path in that will stretch from downtown to the beach, and some communities are more walkable than others as fa as sidewalks, leisure walks and runs... (typically higher income areas and gated communities)
If you live downtown, near the stores and riverwalk, you can absolutely walk to places. There are entire long conversations that can be had about downtown and the various areas... if that is something you are interested in, do alot of research or ask more questions. There are alot of great little local businesses, the farmers market grown every year, and there are new developments and improvements happening every day, but it has its struggles like any downtown of a small city.
Overall, i like Wilmington. I'm from the NY Metropolitan area, so to me its like a big suburban town, but does have a "small city" feel. There is music, theatre, loads of activities for kids, and if you find your niche - there are communities and pockets of people that I have found to be progressive and forward thinking.
With 4 kids there will be alot to do. Look at wilmingtonmommies.com or meetup.com for an idea of local gatherings. The mom and kid community is massive - EVERYONE has babies and kids with more on the way!! I have never see more pregnant people in my life than in the last few years here.
Lots of schools - public, private, and charter, and like any town there are "good" ones and "bad" ones... and events too. Free music in Mayfaire (big outdoor shopping and dining area - google it) every Friday night in the summer, with kids running everywhere and parents enjoying themselves.
THe beach is gorgeous... if you are a beach person, being this close to the ocean is amazing. In the summer, it can be rough with tourists - we dont go much. But now that college kids are back in classes and tourists are gone, its AMAZING! Like being on vacation if you go out in the AM and walk... i would have a hard time leaving it.
Have fun making your decision... let me know is you have and ?'s.