honestly just adding some vintage gold or brass hardware would make a world of a difference

If it's veneer, lightly sand it, then paint it with a good primer and finish up with with a new color paint and hardware.

I second the knobs. Hobby Lobby is a great source for colorful and vintage inspired, too.

http://shop.hobbylobby.com/search/?keyword=knob&match=AND&F_Sort=2&perpage=96

I agree, it looks very promising as is. Just get new drawer handles and it'll be wonderful.

Contrasting paint on the drawers goes a long way. This is a solid wood piece but you could use dark paint instead of the stain.
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http://beechwood1928.wordpress.com/2012/09/09/refurb-before-and-after-dresser/

Paint the drawers in a gradient! Watch the video promo for YHL's new book to see what I mean: http://www.younghouselove.com/about-2/book/

New hardware is a good idea or maybe a wallpaper makeover (just on the drawers)? If its really cheap and you don't want to make it a "forever" piece you could just us wrapping or crafting paper and some modpodge instead of more expensive wallpaper?

Zinsser BIN primer! It's specifically formulated to adhere to slick surfaces (like nasty veneer)--it was a lifesaver when I refinished my veneer cabinetry. Just slap a coat of it on, and then continue with any fabulous color you want. I agree that new hardware will also help--check out Etsy's vintage section for some more cool options.

What applying a graphic decal as a stencil that breaks across the drawers, painting around the decal another color (complimenting your decor), then pulling off the decal to reveal the white below? I'm inheriting some antique dressers (nothing fancy) that need to be refinished, but until I have the time, space and money to do that properly, I'm thinking of applying this type of treatment to them.

Paint the beveled edges of the drawers only. Maybe the legs too.

If you want to test it out, try the decorative tape that I keep seeing on AT.

I second the Zinsser primer - I painted over an IKEA bookshelf (light over dark) after sanding very lightly and applying just 2 coats of primer. It looks great and the paint doesn't chip off at all.

I also second the hardware update...honestly, I don't know why I'm commenting at all since everyone else is right on :). Since you got it cheap, just have fun with it and make it into a piece that you love now.

Depends on how far you want to take it and how comfortable you feel with DIYing. Just replacing the hardware would help. Painting the drawer fronts would be slightly more work but easy enough. A bit more work would be to paint the whole thing, your options there are endless, or you could do a complete overhaul, replacing the top or tiling it. The shape is good so no need to alter that, but here's an example of a complete overhaul I did on a dresser that was part particle board, part pine. It had some dings, so I used wood filler on them, then had my boyfriend (he's better with power saws) change the bottom outline, then sanded, primed and painted. I also had a remnant piece or green marble cut for the top. If you decide to go with a vivid color get an appropriate gray primer, as you'll get better (truer) results.
http://i288.photobucket.com/albums/ll196/rainarana/project-2.jpg
http://i288.photobucket.com/albums/ll196/rainarana/dresser2.jpg

You can purchase O'verlays (thin, laser cut decorative wood pieces) in different sizes, glue/staple them on, then paint the whole thing.

http://www.myoverlays.com/

Decoupage it with wallpaper or fabric.

I'm decoupaging a dresser right now using scrap book paper. It's a problem wrapping the heavy paper around the edges. But use a thinner paper and it could work for you. You can also try contact paper. You can remove it if you don't like it.
You can also glue on mirrors, but it can get expensive.
Try some new hardware first to see if helps.

Trash it. It won't last. The bottoms of the drawers in those cheapie dressers pop out with the slightest bit of weight in them. Trust me, we just got rid of a very similar partical board dresser that was in my daughter's room. When it was just baby clothes, it was fine, but as she grew, it became clear that it was useless. I found a solid wood dresser at a consignment shop, refurbished it & it wound up costing about the same. Inexpensive does not always mean better. Sometimes you'll wind up having to spend more money fixing the cheap thing & trying to make it work.

Just spray paint it, without sanding. Still add some nicer hardware that you like. it has a nice shape!

Stripes.

I say vintage pulls and upholstery tacks! Trim all the edges of each drawer. You could use the ones that come on a roll so it's not as labor intensive...

It would help if we saw more of the room it is in. Doesn't look like it is worth putting much work or money into - keep your eyes open for an inexpensive replacement that is better quality and worth fixing up. This type of furniture does not last and can offgas some unhealthy chemicals.

For those of you who are looking for Melamine Paint it can only be purchased in Canada.

Use Modge Podge and cover the drawers with extra beautiful wrapping paper.