i have had great results staining. i take tempera paint & thin it with water. i tried kool-aid because i know it works for cloth but mine just showed up on the wood like water does. i don't remember which color i used though, i only tried it once.
i've used food coloring & added vinegar to it, just like the easter egg coloring recipe. i didn't want it to rub off when i sealed it, but it still did a little. that was early in my wood staining "career". now i know that i just had too much color on the wood & i needed to wipe it down better. i didn't have good luck with the colors turning out the color i wanted on the wood though. yellow was fabulous if i mixed it with a bit of red and blue NEVER came out blue. i think that had something to do with the wood seeming slightly yellow. and we all know that yellow and blue make green...that's what it always turned out! with the paint though i always got blue & the yellow was just as good as the food coloring one. the only color i use now in the food coloring is red to get pink, it's really great after it's finished! i never bought tempura pink & tried it but i don't do pink very often!
if you use the tempera paint you'll be adding quite a bit of water compared to paint (i use the wet paint, not the powder, but i'm sure the powder would work fine). just as an example: 1 Tbsp of tempura to 3/4 cup water. but keep in mind it will depend on if you want a slight coloring or a brighter one. mix it VERY thoroughly or it will be clumpy and if it sets for a while mix it again.
try testing it before you put it on your toys to be sure you have a good color-it's not going to look on the wood like it does in the dish. the test won't be accurate on treated wood though b/c treated wood has a green tint & that will mix with what you have & come out a different shade.
this is only a stain though so it won't cover the wood grain or blemishes. i like the look of it because you still get the natural "flavor" of the toy when you can see the grain.
keep in mind too that when you put water (the watered down paint or food coloring) on wood it will soak in. when it dries you'll have to sand again because little snags that were sanded off will pop up again. i guess it's the grain expanding with the water.
so, stain, sand, then finish. I finish with water based polyurethane, clear semi-gloss and it looks great. generally i put 2 or 3 coats on depending on how it feels. between finish coats i sand as well with very fine sandpaper. i don't know the grit but is says it's for finishing.
if the toys have wheels that can't be taken off be careful getting the polyurethane down in there because it could drip & then dry, making the wheels stuck.
whew, that was a lot.