I refinished a set of dining room chairs with denatured alcohol and steel wool. Just dip the steel wool into a bowl that has denatured alcohol in it and rub the wool on the wood. Then wipe the wood down with denatured alcohol on a cloth to remove any residue.
You might take a photo of the table as it is now, and take that photo to a hardware dept/hardware store, and tell them what you are aiming for. They will make some suggestions, and then stand there and read labels on different products.
You can go from simply softening the finish and resmoothing it with a finish restorer, to stripping it all the way. There are different approaches. A really good scratch restorer can work wonders. I worked in an antique shop for 9 years and it's amazing what that can do. Often you don't have to restain. I'm trying to think of what it was called, not available in stores, maybe someone out there knows what I'm talking about. Not Formby's, not Watkins, HOWARD's. Yes Howard's has some great, professional products. You'll have to track them down if you decide to try them.
I tend to go the route of buying good sandpaper. I just sand down the furniture to a smooth finish so the new stain has something to adhere to. It has always worked good for me (and I didn't use any chemical stripping agent). Right before you stain the piece, just quickly wipe it down with turpentine. Good luck!
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