I cannot see the grain well, but I agree it probably is not oak. However, it could be birch if it is not old pine. The desk looks nice enough to spend a good amount of time on restoration.

You can unscrew the pulls(from the inside of the drawers) and remove them. The shadow under the pulls will remain and have to be sanded away to match the rest of the fronts.

To properly refinish/protect the wood, you will need to get all the old finish off. That will be a major job, taking weeks to do properly. In fact, that process is so involved you would be better informed by buying a book on furniture stripping and restoration and finding the specific steps needed for that desk.

Once the finish/dirt is removed, the color will be changed. You have several options at that point. Dye, stain, or just finish. Those options will be covered in that restoration book.

If it were mine, I would use a dye(better color control/longer lasting color). But, dyes are expensive and difficult to use. An oil based stain(available at home improvement/hardware/paint stores will be much easier/less expensive.

Then I would use a wiping varnish---NOT polyurethane!!!!
Why? Because that finish is closer to the original(which also could have been shellac), much easier to use than shellac, can be spot repaired(poly cannot be easily spot repaired), and wiping varnish is the easiest finish for a DIYer to apply and get a really nice finish. The wiping varnish takes two applications to get the same coverage/protection as brushed/sprayed on varnish, but allows the applier to avoid runs/sags/etc. that can occur.

>I'm fairly sure it isn't oak...but how can I tell???

Based on the photos, I'd venture a guess that it is pine, particularly with the knot on the middle right drawer.

>The white spots on it in the "open" photo look like paint marks. How do I get rid of the white paint marks without ruining anything?

This looks like paint transfer from scuffing against painted woodwork, such as a door frame. This will easily come off with a little 0000 steel wool, done gently.

>I'm also not crazy about the pull knobs or handles. How can I change them out and not have any problem with the color of the wood behind the handles?

The simplest way is to get replacement with same or larger footprint. Pull them off and have a look underneath, but I'll guess the finish under there is in much better condition.

>What do I do next? Do I put some kind of finish on it to protect it or ? How do I know if it already has a finish? To my untrained eye, it doesn't have a finish because I don't notice anything shiny or satin on the wood...it's kinda dull.

While you could strip and refinish, it might be appropriate to try refurbishing it by a good cleaning.

If you do decide it needs refinishing, I'd agree with Handymac that a good non-poly wiping varnish would be a good choice. You can easily make your own by mixing a brushing varnish with an equal amount of mineral spirits.

However, home center stains, particularly those that are pigmented, don't always work well on pine as it tends to splotch.

Thank you both for the helpful information!!!! It's much appreciated!!

Smiles,
Sooz

I agree it isn't oak. It looks like an Alder or maybe Cherry.