The first step should be to chemically strip the parts to get the finish off and bring it back to bare wood. I use Kleen Strip paint and varnish remover. Then since you have have it apart I would sand the parts through 120 grit paper and assembly it. Then after it assembly, I would sand it to 180 grit paper. The wood is walnut so when you start finishing I would start with a pastewood grain filler. Sherwin Williams makes a good grain filler but it is a natural color. They can add raw umber and red oxide tinting color to it to make it a walnut grain filler. The grain filler is like a thinned wood putty. You brush it on and allow it to thicken and rub it into the grain with a clean cloth rubbing in a circular motion. After drying overnight lightly sand the wood to get the excess off the surface and stain if you wish or use a clear finish over the top. I would recommend Behlen rock hard table top finish for a finish.
Picture loaded after I posted. It appears to be some glue joint problems with the top. You probably should break or cut the bad joints apart and reglue them.