Perhaps the easiest way to be a responsible 4-wheeler owner and operator is to be brought up as one. Since 4-wheelers are relatively new on the scene, it is safe to say most older 4-wheeler operators are adults who already have experience driving other types of vehicles in diverse weather conditions. In addition to these habits of driving, many 4-wheeler riders have not had extensive practice driving on the trail and that's where riding a 4-wheeler differs greatly from street based driving. Most towns and municipalities do not allow 4-wheelers on the paved streets except to cross to get to another trail.
If you have purchased a 4-wheeler, contact your local Bureau of Motor Vehicles and find out if they offer lessons for safe handling on an OHV vehicle. Most do, if you live in a smaller town, check the internet for an ATV or 4-wheeler enthusiast group near you and they should be able to point you in the direction of safety instruction or may even offer classes themselves.
Investing in some safety lessons for your child is a great way to help them learn to ride safely and minimize the risk of accidents. A perfect way to do this would be to invest in a small, kid sized 4-wheeler and allow your child to take lessons. Most instruction courses also give a certificate of completion which some OHV parks are now requiring children under 15 show in order to ride.
Some of the essentials to know about 4-wheelers would be: knowing what and where are the basic moving parts on an 4-wheeler, learn the minimum tools necessary to have when you're out riding, basic maintenance for a 4-wheeler, handling different terrains on your 4-wheeler, balance and rollover points on your 4-wheeler and other safety tips.
Children ultimately learn by seeing and doing. If they watch you as a responsible 4-wheeler owner/operator they will learn quickly to be responsible riders too.