Normally you want a happy bright setup for kids, so a white one is a good start. (I do not like black for kids...)
If you are doing a lot of that try print a theme background. Clouds, a Christmas tree, a farm theme.
And get some props accordingly, some puffy white pillow for the Cloud theme, present boxes for the Christmas background, some cowboy hat for the farm. Something the kids interact with.
This will make a more interesting and stimulating setup for the kid himself (herself).
The background should be printed in a matt material. Some canvas has a glossy side but can be printed on the backside.
Textured backgrounds are a standard for portraits, but personally, I think it's better for teens or kids above 6-7. It is more boring and less cute than the theme background.
But you also can look at the cloth store, in the section of curtains or fabric for furniture.
Forget the green screen. Leave that for video. It is better to have a good photo straight from the camera.
You do not want to photograph kids in the floor. So construct a really sturdy table of a good size, where the parents can look for the kid on one side of the table.
Some EVA foam pads to make it comfortable and prepare some cloth to cover it with diferent colors accordingly to the main setup (easily washable too). Some elastic ones with matt finish are better, like lycra.
Yeap an interesting question is this: What are the differences between different types of cloth backdrops? regarding textures and wrinkles.