Use a hard, inexpensive local wood, without large pores. Hard, since they will get banged around. Pores will allow the wood to trap bacteria, dirt where you don't want it.
Maple is my first choice. Second choice might be birch, sycamore, or alder, if you have a source for them. Poplar is not bad either - cheap, reasonably tough. Cherry would work too. Much depends on what you can find locally. Do NOT go to your local big box store (Home Depot, etc.) to buy hardwood. Hardwood bought there will be quite expensive for what you get.
Woods that I would NOT use are imported exotic woods. These can often cause allergic reactions. Another wood to be avoided for this purpose is walnut, which might cause reactions in some.
Use wood coloring, rather than paint. Paint can chip off, and may not be food safe. And be careful with stains, since many of them will not be food safe. Wood dyes, like aniline dyes, may also be something I would not put in my child's mouth. For a coloring, I would use simple food coloring.
Pine is a difficult wood to stain under the best of situations. Pine has areas of different density, color, and resin (pitch) content. Using a product like Minwax prestain helps a bit. Follow the timing instructions carefully when using a pre-treatment. Little tips: i like to apply a liberal coat of oil based stain with a sponge brush. Let it set a few minutes, then rub it in and evenly distribute with a lint free rag. As one rag becomes saturated with stain, I use a second drier rag to finish the rub in. The wetter rag is great for distributing the color. You don't want any really wet stain remaining on your wood after the rub in. Once you get to the urethane, sand lightly between each coat with 220 grit paper to remove any dust bumps. Remove all sanding dust with a slightly damp cloth, let it dry and apply next coat. Again, I prefer a good grade sponge brush, but a good grade regular brush for oil/urethane works good for appling the urethane. Remember, thin coats, and sand between each coat.
Best Answer
Most modern enamels (aka "oil paints" or "alkyd paints") cure fully in 14 days, at least according to the product data sheet provided for the Sherwin Williams paint I just used for my friend's kitchen cabinets. Either way, an oil or water based enamel will provide a much more durable and resilient finish than a latex will.
I would suggest sanding the wood with 220 grit sandpaper, applying a high quality primer compatible with the final finish that you choose, and then a high quality water-based or oil-based enamel paint. I've really liked the results I've gotten with Sherwin Williams Interior Water-Based Alkyd and Benjamin Moore Advance Water-borne Enamel paints. They are true enamels, but clean up with water. Again, be sure to use a compatible primer. According to the attached technical data sheets, the waterborne alkyd paints take 30 days to reach their full hardness.