The sides and front might be wood but the top looks man made. Can you confirm material? With that nailed down we can make better suggestions.
Meanwhile I will make general statments aimed at beginners. Rejuvinating origional finish if possible is much preferred over outright refinishing. There are several options for freshening up an existing finish depending on what the existing finish is and condition of course.
Wherther you want to refinish or touch up,figuring out what's on the wood is very helpful in achiving good results and often is the difference in complete disaster.
A thourgh cleaning is nessary before undertaking either approach and often reveals the finish is in servicable condition without going farthur.

It would also be helpful to know where you want to end up. Radical change in color, touch up only, swap out the hardware, paint, shabby chic, etc.

At the risk of sounding like an idiot, how do I determine for sure what the materials are?

The bottom is wood for sure; when I turn it over it's unfinished and there are a couple of nicks (probably from a vaccuum) that show bare wood.

The top isn't wood; it's super shiny, I'm guessing laminate?

The sides and front are...well, I don't know! They're shiny, and I sanded a teeny tiny little spot on the side that will be against a wall, and the shinyness came off. (My gosh I sound like an idiot, don't I?)

I don't like the brown color or the hardware. I plan on getting two new knobs, which is the easy part. Ideally I would like to have it be a mahogany color, which is what the rest of my furniture in my house is. If I have to paint the top piece (since it's laminate) but can stain the rest, I could live with that, but I really want the rest of hit to be stained and not painted.

Thanks for any input you can provide.

Here is a general primer JJ,books have been written that still leave unanswered questions so keep that in mind and just do the best you can. Spray or sprinkle water on the top to see if it beads. If it does,clean with specialized SILICON REMOVAL product fron the paint store. Do the same on front and sides. If water doesn't bead,wipe the surface with naptha,let dry and sand with 120 grit garnet sandpaper until when you wet a spot with water then emediatly wipe the surface with dry cloth the surface remains damp then eventualy air dries. That test tells you that top coat finish has been sanded away and stain will penitrate wood. Before applying stain wipe off sanding dust with a tack cloth or cotton cloth dampened with thinner that is reccommended to clean stain from brush. From here it's simply a matter of following directions on container to the letter. I suggest you pick up a color wheel from and arttist's supply store and study up on how to use it. That simple little tool will put you light years ahead of most diy painters and decorators. Once you understand the wheel you will refer to it many times in coming years while selecting decorator color combos. What you will use it for here is to select the correct stain color to apply over the existing color to achive the desired resulting color. After a sample stain has dried dont overlook to wet a spot with paint thinner,saliva or the topcoat you plan to use so you can judge the depth of color before doing intire piece. Needles to say,experimenting on a concealed area while trial testing color is helpful.
I have good luck getting paint to adheare on laminate surfaces useing primer from big box reccomended for pvc.