Steph, if you haven't tackled this project yet, might I recommend using a spray primer and then spray paint.  I've painted just about every form of wood there is out there and I always spray paint when the color I'm looking for is available in a spray.  Not need to seal first.  The primer will seal it too. 

Thanks for the advice! I think I'll give it a shot this weekend ... and will definitely post the finished product.

Your drawer bottom looks like MDF or masonite. That stuff will soak up a ton of liquid (paint). You can buy sealers for wood so you are not putting a gallon of paint on something that would normally take a quart or less. You may try a sealer, primer, then paint. Find a woodworking expert to ask because I think you can get all of those products in spray form to avoid primer brushmarks under your new spray paint. Also, with the sealer you may be able to skip the primer step. Good luck and dont forget to post the finished result.

I second heylucy's comment. Prime then spray paint to make it easier on yourself. When you're done with that, you could follow up by painting the front edges (the edges facing up in your pic) a different, contrasting color by brush. That could look cool!

I have a similar shelf thingy. I've managed to get it primed, but it was kind of a pain with all the tiny little compartments. I'm thinking for the next coat I'm going to try spray paint. So those are my suggestions, I would prime it, and then use spray paint for color!