If you know people who live in the new city, ask them. Describe what you like and see what they have to say, maybe say you're looking for "a small town in the city." Also, visit the prospective neighborhoods and sit in a public place (like a park, coffee shop, etc.). Are there people out and about? Do they seem to know each other? Are people out walking their dogs (great way to meet folks)? What kinds of fliers are hanging up on bulletin boards--are there plenty of local activities going on? Go hang out at the school when it lets out--are there parents standing around talking in groups? Are there lots of people calling out "hi" to other people? Is there a neighborhood newspaper or newsletter? Read it it if there is? How do they celebrate the Fourth? National Night Out? Also, look for a neighborhood with sidewalks, no fences in the front yard, front porches that are open and have furniture on them.
I live in such a neighborhood and count my lucky stars that we stumbled upon it as starving grad students--we started in an apartment, then bought a small house, then a larger house when we had kids. I'd like to live in the neighborhood forever, downsizing as it makes sense.