Finding the right bike for a child is a lot simpler than choosing one for an adult. There are really only three (3) parameters:
Bicycles for the smallest children are really sit-down scooters.
more↴
On some of these bikes, children can put their feet up. This helps them develop and a sense balance and go a little faster.
more↴
Training wheels are often the next step. They allow the child to focus on the tasks of pedaling and braking, without having to worry about balance.
At this stage, the child must be aware of where s/he is, anticipate where s/he will be and maintain awareness of her/his surroundings.
Children's bikes vary in both frame AND wheel size. As the child matures, it is important that she have a properly fitted bike that is right for her height.
The child should be able to reach the handlebars comfortably without being overextended or cramped. Remember that the seat can be raised over time.
A bike that is too big or too small will be awkward to ride and difficult to control.
The type of bike, the question of gears or no gears, etc. is a function of how and where the child is using the bike, not her age.
more↴
As with adult bikes, consider how the child will be using the bike:
Finally, you may wish to buy a bike that the child will grow into they'll grow out of it soon enough.