I have two kids and a wife. I'm the only one who rides a bicycle with any proficiency. Just thinking about them trying to ride along with me makes me feel ill. After the tenth panic attack I'd probably collapse.
I meet them at the mall/big-box store/market and haul home whatever is purchased.
They get back on the bus and come home.
I usually arrive before them, we keep in touch and coordinate using cell-phones.
If you're living in a rural area use your car to haul your kids and their equipment to a park with a bike/fitness path and have lotsa fun.
You know the deal. Going car-free isn't going to happen for you unless you're planning to move into a city.