I think the easiest approach would be to use General Finishes Gel stain. They have many colors to chose from but I'm not sure if they offer one that matches your furniture. It covers like paint and requires little prep work. I used the Java on my kitchen cabinets and they look great.

Here is a link that might be useful: General Finishes

I'd like to see a photo too before I suggest anything.

It sounds like cherry and may take a series of stains to make it more "brown". but, agree that picture is needed.

OK I'll get a picture tomorrow. It's too dark in the basement to get anything more than a reflection of my flash tonight. So after work tomorrow afternoon I'll see what I can do. Thanks.

OK here are some pictures. What we have done so far is use a stripper on the top, let sit for 20 minutes, scraped lots of gunky stuff off, then repeated. Next we used mineral spirits to clean up the remaining residue. Then we sanded the top. We really haven't adequately sanded it because we ran out of sandpaper so the sanding is quite spotty. I then cleaned up again with mineral spirits so that I could take the pictures. We haven't done anything to the rest of the piece. It is incredibly red. The back has some markings on it including one that says, "mah." I assume that means mahogany, but I don't know if that refers to the wood or the original finish. I really don't want this thing to be red. I need it to be light to medium brown. If I have to go to a dark brown, then I might as well paint it instead of staining it. Any suggestions?

It looks like mahogany to me. Instead of trying to try to stain it to match the existing furniture, which I think will be a challenge, why not paint it and give it a shabby chic look? It will set it apart as a special piece instead of having it look like it "almost matches". If you post a picture of your DD's room, I'm sure you'll get several suggestions what would be a good combination.

If it's mahogany ( not sure - part of it looks like it could be but not all) and you decide to paint it, you will need to use a sealing primer first or an isolating shellac coat before you paint or the red will bleed through any paint you use. If you use white paint, you'll have pink streaks. Even if it's not mahogany, I'd still seal it before painting to keep the red in the old stain from bleeding through - just to be sure.

To me it looks like cherry and can even be pine. Here's a test: mahogany is a very hard wood, if you can dig your fingernail into it, it's not mahogany. That said, to get to bare wood, you have a lot of work to do. And to get it a color that is not "red" will take a series of brown stains. It's possible, but not an over-night project and not in one step.
If you choose to paint, you will certainly need to prime with a stain blocker BIN sealer or KILZ. You will have to finish removing the "gunk" from the top and do a light sanding to the rest. Prep will be key to a decent paint job.
If you don't want the work of stripping the entire piece, or don't want the look of just a color, there are a few techniques to consider.
An antique finish, where you paint and then sand the areas to reveal the wood underneath. Or, shabby chic (as mentioned above).
Or, you can paint it to look like wood, "faux bois"

It's hard so I had already ruled out pine. Do you think I need to do a couple more rounds of stripper or should I just proceed with sanding at this point? Remember I've only worked on the top so far. I haven't done anything to the rest of it. I'm only working on it a little at a time.

One more coat of stripper should remove the rest of the yucky stuff and then do the sanding. Use different grits of paper, starting with rough and finishing with fine.

Have you thought about replacing the drawer pulls? When you get to the drawers, I'd just remove what you have now and replace them--that way you don't need to worry about refinishing the pulls.