When i bought this desk that my computor sits on,it looked like it was stained and varnished.But someone had set wet glasses on it and it had those white rings that wet glasses leave.I never gave it a second thought.I got out the sander,sanded it,applied new stain and then varnished it.The drawers were in good shape,they were painted with turtles stenciled on them.I left them as is.Since i've had it i kinda messed up one of the drawers,by forgetting to close it,and then my chair scraped the paint.I have to see if i can find a paint that color green for touch up.

So,having said all that,i'd say sand and restain and varnish.Good luck.
Kathi

The type of finish you want on a piece of furniture sort of determines how you prepare it. If you have a painted piece that is in good condition then a good sanding and then painting is all you need. If you are putting a latex based paint over what might be oil paint , then you should put on primer first. But if there are many layers of paint and it is chipped and flaking then you're probably better to strip it all off with a furniture stripper and then re-paint. With pieces that are stained and varnished and are going to be stained again, I strip everything off with stripper, sand, stain and varnish. If you are going to paint over a previously stained piece and the finish is in good shape, a good sanding then a primer and then your final paint should work. If you paint right over stain the stain might bleed through. Finishing furniture is like a lot of other things-the prep time takes a lot longer than the final finishing.
Try your local library for books on refinishing furniture. You'll get lots of info there.