Hi Pookietooth, I explained part of it above. We live in a very small town where word-of-mouth networking happens almost without any effort. There's a newsletter where you can put out notices about meetings and get-togethers, even just "we're going on Sunday afternoon hikes, and welcome anyone who wants to come along" sort of things. And even though I'm pretty introverted, I found it easy enough to organize stuff here. I started a music program for young children (I'm a violinist) and then a gardening/environmental club for families with children. Now, part of my success in organizing stuff came from the fact that I did have a social network already going, so a fair number of people knew me and were happy to support stuff I was organizing. I didn't have any really close friends at the time, but I did have a fair number of contacts.
To establish that social network I started volunteering with local non-profit organizations and at the local school (library stuff, helping organize the arts-and-culture festival week, etc.). I prefer low-key, behind-the-scenes work, for the most part, like setting up chairs before and after community events, baking for receptions and open houses, cleaning up after choir concerts, doing litter-picking by the highway, but gradually I got known as a decent community-minded person. A lot of that volunteer work I was able to bring along and include my kids in, which was a bonus, because people got to know them as well, and that really developed their social networks too. Yesterday, for example, my 9-year-old worked as a volunteer number-caller at a big mountain trail race near here, radioing in the numbers of approaching finishers to allow the announcer time to cross-reference stats, names and locations before doing the finish-line announcing. It was a real, important job, more responsibility than most 9-year-olds would be granted, but my dd has been so busy at community events over the years that people trust her with these sorts of jobs.
So I guess I would highly recommend volunteer work as a way to involve yourself in your community, help make it a better place, and begin to develop a social network, even just of acquaintances, on which to draw for possible friendship and support. For both you and your children.