For my kids, they started with a 2006 Yamaha PW50 at 5 years old. This is an awesome indesctructible bike. Mine sucked some dirt and scarred the piston/cylinder and I am not mechanically inclined, but I changed it myself.
Next was a 2003 Yamaha TTR 90. This is again in the lawnmower category. Great beginner bike, with an auto clutch and 3 gears. Very capable bike. Still puts a smile on my face.
Racing started on a 2004 KTM 65 SX. This is a different bike from the other two. It has fully adjustable suspension, a 2 stroke motor and hydraulic clutch. It can be used as either a 'normal' bike up to half revs and as a crazy racer above half revs :)
Kid handled racing pretty well and will continue with race bikes, an 85 and then 125 etc.
Next: Dirt Bike Setup