I still run quite a lot, but with the popularity of "multi sport" (triathlons, etc) plus the draw of being able to go farther and bring more, we have borrowed the concept of the road train! Certainly I am not the first parent to string together several wheeled devices, but we love it. We sometimes go to the grocery store or to cub scouts with all in tow. Even with several young children at home, you could train for a cycling event! Probably not a century ride, but these kids would be cool with 10 miles. Revisit Destination Running to break up the distance.
You may be able to tell that I have all this fun stuff hooked up to my single speed bicycle. That means that there are no gears. If you wanted to try a trailer bike, a trailer, or both, I would recommend a geared bike and more specifically a mountain bike or hybrid bike because those types of bikes have an easier gear ratio than a road bike. Or do what I do if you just like things crazy hard :)
I've been running over 1000 miles a year with my jogging stroller for the last decade and more recently added in biking with a trailer and trailer bike. I've collected some fun ideas to bribe- I mean encourage - kids, as well as mechanical advice for maintaining your equipment.