It's important to always check second hand furniture from craigslist, garage sales and second hand stores before you put the furniture in your car. You want to be sure there are no creepy crawlies underneath or in the drawers. I used to live in California and trust me, you don't want a black widow coming home with you! There was one time I was at a garage sale and saw a guy start putting a chair in the back of his car only to realize there was a gigantic black widow living underneath it. Once you get home just give the piece a good clean, wipe it down with a mild non-toxic cleaner and if there are drawers then give them a vacuum.
Now I know the pros would always recommend stripping old paint, sanding and so on... But honestly when you're refinishing a piece of second hand furniture that you bought for $10 and you know your own kids are going to bump and scratch it, it's okay to cheat a little. It just makes sense to keep things simple when it comes to refinishing kids furniture. If it's not an antique from grandma, and not going to get a lot of "traffic" in the home then I recommend keeping it easy (and cheap). The "furniture restoration police" will not come knocking at your door if you don't do it like the pros. It's really OK.
I do however recommend a quality primer. I gave this piece two quick coats of a plain white bonding primer. Then I used blue latex paint on the front of each drawer. I only needed to do two coats on each drawer. For a project like this all you need are the small paint sample sizes from the hardware store. If you just pull the swatches you want, pick as many colors as you need (depending on how many drawers you have) and then just have an assistant at the store mix you up the small sample sizes.
For some reason I ended up with two different brands of of paint samples. I think they were out of a certain color in one of the brands at the time, so we combined two brands (Olympic and Valspar). It all worked out. Mark the top of each sample container in the order of the drawers - with 1 being the top drawer and so on. Please note - my pictures are all taken with my iPhone, and it appears that the two middle drawers are almost identical. In person its a little more obvious that they are actually different. But the difference between those two drawers is still a little too subtle, so you may want to try a more dramatic difference in shades if doing a similar project.
Once the paint has dried it is a good idea to give it a quick seal/finish. Two coats of Polycrylic works great. Since this project we have moved twice (one of those moves was from California to Utah) so the chest of drawers has been knocked around a lot and there are a few small scratches on the drawers that I need to fix. Overall it's held up really well with moving and general wear and tear.
Finally adding new hardware is always a great idea. I chose some cheap (but fun) shiny silver knobs from the Home Depot. I thought the shiny ones added a little fun pop, perfect for a kids bedroom.