A child belongs either in a side car rig attached to a motorcycle or in the passenger seat behind you.  Never, ever place a child on the gas tank.   

Kiddie riding rules:

Feet belong on the pegs, never stand up on the pegs while riding.
Sit straight up and hold on tight, don't wiggle around.  
No removing garments, helmets, glasses, etc, or playing with items that can fly away while riding.

Kiddie talk:

No. Not baby talk!  Create a signal for potty so the child doesn't get crazy back there trying to hold it in.  The same goes for if the child gets sleepy because they will slump over.

Kid's Footpegs:

Please make sure your child can reach the footpegs.  NO dangling feet ever.  There are all kinds of footpeg assemblies if the little ones cannot yet reach the adult passenger pegs.  Inquire within the pages of the aftermarket catalogs or ask your local motorcycle shop. 

Seats:

The passenger seat should have a passenger strap as well as a backrest for the child.  

Kiddie Riding Belts:

Some folks like these, some don't.  But, a buddy belt provides additional safety confidence if your child unpredictably falls asleep behind you and it's a good training device for their first time passenger experience.  It straps the passenger to the rider.  Some popular ones are:

Buddy Belt 

Child's Riding Belt 

Kiddie Gear:

Just like adults, kids need riding gear.   There are many child sizes available in Motocross gear.  It's imperative that the child wear a helmet that fits well preferably with a face shield.  Gloves, boots above the ankle, a sturdy jacket and chaps, too.  Make sure that the jacket is a riding jacket so it doesn't balloon on them with cold wind.  Also, ski pants are good for the kids to wear, keeps them warm, too, as long as they are not too baggie.