I would start with an area rug on top of carpet and then pull colours from the area rug for throws , pillows etc....

Some inspiration, love the wallpaper, MOON PHASE WALLPAPER (SELF-ADHESIVE)

I would like to know more.

1. Is this a playroom, or is it part of a living or family room? I wouldn't mind more pics.
2. How old are the kids?
3. Are you able to buy new/add new pieces?

Off the top of my head:
Large colorful play rug like day care centers have (map, a road)
Pillows in colors from the rug (red, yellow, orange)
Built-in shelving/cabinets on the wall where cubbies are--or a non tipping cheaper alternative with repurposed furniture
Craft table with 2 + chairs, bright color
Picture ledges above couch etc for brightly colored art
An area to run and play like nerf basketball or hockey
Large baskets for bigger toys and stuffed animals

Pros, how am I doing so far?? Keep us posted!

All of the above.