If you are going to do mostly moto-style riding, but want to do occasional off-roading, I would consider the KTM 200/250 MXC first, because they have the lighting coil you are looking for. For the kids, if they have never ridden, an XR 50/70 would be an excellent choice.
There are a lot of bikes that will work for what you want, both for you and the kids. Your first consideration should be getting a good deal on a bike that is a brand you can get good local support from a dealer or shop from, don't just decide on a particular brand to the exclusion of all others. Any good condition125/ 250 two-stroke MX bike can make a good offroad mount with minimal work, and people even moto KDXs and XRs.