You do not have to be competitive cyclist to go for a cycling vacation. Cycling is not only a great way to keep fit, but is a much more environment friendly approach to moving around. It is also one of those activities that every member of the family can participate in.
When cycling, you have more flexibility in quickly exploring the areas you go to. This is unlike when you travel by car in which case you may have to abandon the car at some point if you want to venture through narrow trails.
There is quite a bit of planning that goes into readying for a cycling vacation. The kids will probably be one of the biggest factors. How old are they? The older they are the greater the number of options you have.
Age and fitness will determine the distance you cycle and the terrain (e.g. rough verses paved road or hilly verses flat). If the kids are too young, then going for a full-scale cycling holiday is off the table. The best you can do in that case is to have the cycling as just a small part of a wider range of activities you embark on during your holiday.
As far as cycling gear goes, the bare acceptable minimum is a bicycle and a helmet. The need for additional equipment will depend on your unique circumstances. For instance, if you want your under 5 kids to come along, you will need a bicycle trailer that you can tag to your bicycle and pull.