Seeking LED advice for toddler "busy board"
For light, you want to use red, green, and blue, due to colour magic. You can buy LED packages that contain these in one piece of plastic. Something like this.

Setting brightness is just picking the right resistor value. More resistance means less current flows, so less light. You could also use a potentiometer (in series with a resistor too) to allow variable amounts of each colour.

For power, I'd probably use 3xAA batteries in series.