I painted kitchen cabinets before, so your bathroom vanity would be the same process.
First of all, you need to sand the item to get the gloss of it. Use a medium grit sandpaper, and finish up with a fine grit sandpaper. (Do this by hand, no sander necessary for this small of a project.) This will help the new finish adhere better. You don't have to sand all of the finish off, just the top layer of gloss. After sanding, use a tack cloth to thoroughly remove any sanding dust.
Next step, prime it with a interior primer. Look for one that is good for wood, not just drywall. Take the doors off first, and do those laying flat. Be careful of drips on the opposite side. Let dry overnight.
Go to Lowe's or Home Depot and look for cabinet paint. It should come in a can just like regular paint. It will be either a semi-gloss or gloss finish. You can get it tinted to the color you want. Do a first coat. Wait overnight, and do a second coat.