How can I completely refurbish this old chair?

I love woodworking, but absolutely hate sanding unless it's the final pass. On antiques, you want to be especially careful with the sandpaper.

Since you haven't specified, I'll make an assumption that the legs are lacquered or varnished with a clear over some wood stain. If that's the case, these steps have always produced great results for me:

Get a small can of acetone and/or a small can of MEK. They both accomplish the same task, but the Acetone evaporates extremely fast, so is good for final cleanup. These chemicals are harsh on the eyes/skin, so wear proper protection. It's the same stuff used in paint remover, without the messy goo and extra $.

You'll also need a couple of green scotch-brite pads, some lint-free rags (old t-shirts), and an old toothbrush.

Dip the toothbrush in the MEK and start gently scrubbing on the wood. You will see the old finish dissolve quickly. Every so often, wipe of the dissolved finish with the towels and use the scotch-brite pads to get off any tough spots or larger flat areas.

Finish up with a final wash of MEK or Acetone. You will notice that the chemicals will also dissolve the stain a bit, which is terrific, since it ends up spreading around to even out any nicks, scratches, and faded spots.

Finish off with a coat of clear varnish, choosing the gloss level to your liking and send it off to the upholsterer.

Good luck!

P.S. Be sure to dispose of your old rags appropriately according to the label directions - spontaneous combustion from a closed garbage can in your garage would not be a pleasant experience.