This will depend on several factors: skill level, time, and money. I have never properly refinished a piece of furniture, the closest I came was over here:
Singer 201 in a No 48 Cabinet - Quiltingboard Forums
This cabinet had some raised joints on the veneer, and a few small spots where it was missing. I've been told they make veneer repair kits, but also that the veneer of yesteryear is quite a bit thicker than that used today. I used carpenter's wood glue to fix the problem areas; the excess can be wiped off (while still uncured) with a damp rag; I put wax paper wax side to the glue on the glued area and then clamped some boards to flatten it out. On the table in my post, in the "after" pictures, you can see where the two leaves join together, a small spot of missing veneer. Because of deep ring marks in this one, I used a pad sander on the whole surface, including sanding down the edges of the missing veneer. You could sand just that area, but it can remove some color.
I typically use Howards Restore A Finish, and did so on that No 48 cabinet. The color I had was darker than the original, but I was fine with that. Howards comes in a wide variety of colors. On 90% of the sewing machine furniture I've worked on I have used only Howards, applied according to the directions. Depending on how nice the cabinet or table is I will follow up with some clear coat, usually just on the top and typically spray lacquer, which has to be used outside as it's pretty nasty. I am describing what I call Lazy Man's Refinishing. But it's worked well enough for my standards. I have also used the Howards on bentwood cases and original portable bases as well.