I bought a table with ceramic tile at a garage sale and want to redo the tile. What is the best way to remove the old tile?

What kind of table is it? Knowing that would help us to help you better here.

I agree we need to know what base material was used under the tile. Basically you need to chip away the grout around 3 edges of one tile near the center, pry under the center strip of grout and once the tile pops you can hammer a putty knife blade under the other tiles. You will need to chip and scrap most of the adesive off the base material, its likely adhesive rather than thinset.