Either food coloring or gel colors (such as those used for cake decorating - I tried Wilton gel colors, since that's what I had) can be used. However, you can't really "paint" with them, as far as I can tell. They bleed tremendously through the grain, so while you could get perhaps some stripes or gradations, any kind of real detail will look a mess.
Additionally, the gel type of food colors must be diluted significantly. If you use too much gel and not enough water, it can sit on top of the wood and be quite tricky to get completely off. I discovered this first-hand. I had to wash the piece, and it's possible it will continue to bleed some color.
However, if what you are going for is bright, solid colors, food coloring works great and can be easily sealed with natural polish or shellac (see page about sealers.)
Update 12/2014: A visitor writes in with the following food dye-related experience:
The food dyes gave lovely brilliant colours on naturally very light plywood. I used them undiluted. Except for the purple, which I was unable to obtain by mixing different quantities of red and blue, so I bought some, which happened to be a gel, so I had to mix with water. The woodgrain raised only ever so lightly, but I sanded very lightle with 244 sandpaper anyway. And I had purchased some beeswax/jojoba oil mixture, which I rubbed in. This mixed with the dye on the toys, so gave me lovely coloured hands, but which washed off relatively easily with normal hand soap. The resulting toys were beautiful, I was really pleased with myself, but when I gave them to my grandson for his first birthday, it turned out that the food dye/bees wax mixture melts on warm children's lips and gives them brightly coloured lips, plus stains in everything they touched afterwards and the toys don't look so nice anymore. I've taken them home and hope a few more coats with the beeswax/jojoba oil will reduce the staining problem.
These I would truly love to try, but it's $17 for 3.5oz of one color - another prohibitively expensive option. You can buy them from Highland Woodworking if you have deeper pockets than I do.
If you're in Australia, a super option might be UBeaut Water Dyes. I wrote to them to see if they export to the U.S., and they suggested food coloring (which as noted is not an entirely satisfactory solution), so I'm not sure if they would ship overseas (their order form is a bit prehistoric.) I may give it a shot if I can't find a really good local option.