If the table is in good structural condition, then simply refinish it to your liking. What you decide to do to it will require different processes.
If you want to see the wood, then sand it until it is clean fresh wood, then stain and/or clear coat. Minwax liquid stains are great, McKloskies polyurethanes are top notch and easy to work with.
If you want to paint, you won't have to sand quite so much because the paint is going to hide stains and such. Sand everything smooth, prime it with a good quality primer, such as Bullseye 123, then topcoat with an alkyd paint.
I definitely want to see the wood. Can I sand by hand? Also should I give any attention to in between the tiles? Would I need to sand between coats of stain or varnish?