Kids Motorcycles / Motorbikes - Electric and Gas
Kids motorcycles are not the easiest things to find. Because children can not have a official driving license, the motorcycles generally are not road legal either so the market is small, and therefore the price is high.
A great way of getting around this is looking into mini motorcycles. These motorcycles use tiny engines and frames and you can actually get road legal versions complying with emission laws and such like.
Original mini motorcycles are authentic replica's of their bigger counterparts. Being a Japanese idea, the motorbikes available were mostly Japanese sports bikes, rather than big America Harley Davidson touring types of motorcycles.
Newer mini-motorcycles are cheaper versions now and made for fun rather than as a serious small alternative to a motorbike. So tend to come from China and look the same as one another just in different colors and sporting different decals. These are still well worth a look for your child though if you are not looking to spend a lot of money.
Dedicated Kids Motorcycles
The childrens motorcycle you are after will depend on the child's age. There are electric and gasoline options available. The gas engines will be geared slowly and are unlikely to be over 50cc. I have seen a few 90cc ones but they are rare and more for a child competing in racing competitions.
A 50cc powered motorcycle will not take off like a rocket and scare your child, but allow them to use it slowly and learn the handling characteristics of a basic upright motorcycle. They won't be getting their knee down on the corners! Many come with stablizers too if you're looking at a younger age range. The bigger motorcycle manufacturers will oblige to providing the younguns with what they need.
Yamaha and Honda make many engines at different sizes so they are in the best position to provide a genuine authentic motorcycle before stepping up to something more serious. They usually have automatic gear boxes too. Stepping up from 50cc is usually greeted with 90cc, 125cc and 175cc engine options. These are usually mated with larger motorcycles designed for larger (older) kids respectively. There is not many options after this apart from the odd 250cc racing bike sneaking through. You kid is probably more suited to a small adult motorcycle when getting up to this range though!
The style of these will likely be dirt bike/scramblers as road going motorcycles have limited use when you're young and muddy and prone to falling off. In addition to these are electric powered kids motorcycles, these are aimed at younger children, don't have the threatening noise, however they have less power (required for off road use) and will need to be charged before use.
Electric powered motorcycles deliver torque instantly and are controlled by the voltage to the motor. This is easier for a beginner as it is easy to control, but your control options over the vehicle are more limited. They are typically used for younger children and many come with stablizers. Think normal push bicycle but with electric motor strapped on the side.
However some nice electric motorbikes are coming through that are bigger, but as there are currently no mass produced electric motorcycle available for adults, they tend to be phased away from children as they get older and demand a bigger bike too.