I can not stress this enough...if you don't take this process seriously and REALLY get rid of all the old finish, you are wasting your time...because you WILL be disappointed. And while this stripping process may seem labor intensive, it is much easier than sanding an old finish off and you are much less likely to damage the piece...which brings me to rule #...whatever, I lost track!
SAND AS LITTLE AS POSSIBLE. You should almost always use chemical strippers to remove ALL the old finish and only sand a tad (with the grain...if needed...220 grit) AFTER you have stripped all the old finish. Many old pieces (unlike this table) may have veneer...veneer can be thin and if you sand too vigorously you can do a lot of damage! Also, you will NEVER get all the old finish out of decorative carvings or turned legs (like this table) with sandpaper...not happening!
Also, palm sanders tend to leave little "swirly" marks...and as faint as they may seem, they are going to be glaring after a stain and finish is applied...save yourself the trouble...hand sand with 220 grit paper and steel wool on the ornate areas. SAND WITH THE GRAIN!! Cross grain marks WILL show up in your final finish!
Keep in mind too that once you strip all the old finish, you are REALLY going to see the "character" (grain) of the wood. A lot of old finishes were applied with a "multi-layer" process or were sprayed on and the stain may have been mixed in with the finish...they did this to give the piece a more "uniform" look and hide the "variations" in the wood grain.
If you like a uniform look, paint it because you are going to reveal the "character" of the wood when you strip it down to bare wood!
Also, there is a chance you could dissolve adhesives (glue) that was used on joints...that's not a huge issue...it is best to go ahead and strip and stain the piece BEFORE you do any major repairs...stain will not penetrate glue...so if you get any glue on the wood, it won't stain! (If you are going to paint a piece repair it BEFORE paint...if you are going to stain, repair it AFTER stain)
Okay, I have now spent 4 hours scrubbing off all the old finish (that is how long it took me to do this table!)
Now, wash the entire piece with clean rags and mineral spirits. I have a tenant who works for a commercial laundry service and she brings me old towels I use as shop rags! If you are going to refinish furniture, buy towels and rags at garage sales and thrift stores and cut them up into washcloth size pieces.
Which reminds me...WORD OF WARNING!!!! DO NOT THROW RAGS OR STEEL WOOL IN A TRASH CAN IF THEY HAVE ANY CHEMICALS ON THEM....LAY THEM OUT TO DRY IN THE SUN AND ONLY BAG THEM UP AFTER THEY ARE COMPLETELY DRY AND THEY HAVE AIRED OUT FOR A FEW DAYS! THEY CAN SPONTANEOUSLY COMBUST AND BURN YOUR HOUSE DOWN!!!! (Do not bag them with paper or cardboard and keep them separate from household trash!)
While washing it down with the mineral spirits you may find a few spots that need a little more "scrubbing" to remove the old finish...not a problem...just scrub it, then wash it down with the mineral spirits!