Hi! I grew up in New Bern and my parents still live there. I think it has good points and bad points, like most cities.
The downtown/river area is BEAUTIFUL. There are some other areas that are very nice as well. The rest is pretty much like any small city. You're only about 45 minutes from Atlantic Beach, and if you like sailing you're only about 35 minutes to Oriental. The political landscape in New Bern is interesting. It is a fairly conservative area, and as you get farther out of the city limits it tends to get more conservative. That said, there is also a decent progressive contingent, too. But, I still always felt very out of place there as a very liberal girl in New Bern High School. I remember being told I was going to hell, because the teacher asked who believed in evolution and only three of us raised our hands. Of course my time in highschool was almost 15 years ago, so things may have changed. What kinds of things do your kids like to do? There isn't a WHOLE lot for teenagers to do in New Bern. Movie theatre (though, I have my own personal issues that will prevent me from ever stepping foot inside the main theatre again). Skating rink. Beach. Mall (which, when I was there was CRAP. But they're remodling it now, so it's getting better). Walking around downtown. Historical stuff. Getting into trouble. I don't say this to dissuade you, because I don't know you or your kids, so that may be all you need in a town. But if you and your kids want to go out late into the night or have something new to do every day, it may not be the place for you. My parents obviously love it, as do many other people I know who live there. The historical aspects were a big draw to me, and still are. You're not too far from Raleigh, so you can make frequent day trips up this way (I'm in Durham), and the triangle area, and both Greenville and Jacksonville are where people tend to go to do bigger shopping trips. There's a fair amount of diversity. Let me know if you have any specific questions. (and sorry for the lack of paragraph breaks, for some reason when I hit enter it's not doing anything!)