Thoroughly clean and dry the countertop.
Put on rubber gloves.
Repair chips with granite epoxy resin.
Use acetone to clean any hard, sticky dried substances like glue or epoxy or to remove ink marks.
Remove stains with non-citrus-based liquid dish soap and a steel wool pad and/or with a paste made from liquid dish soap and flour. Soak tough stains by covering them with the paste and letting that sit for several hours. Hydrogen peroxide and ammonia are also helpful in removing stains and can be used in the paste instead of dish soap.
Buff deep scratches down by lightly scouring them with a fine abrasion, diamond particle buff pad. An electric, orbital buffer can be used. Work the area until it looks polished again.
Apply non-abrasive granite polish and buff the counter with a soft, clean cloth.
Seal and shine the countertop with a solution of either rubbing alcohol and a few drops of non-citrus-based liquid dish soap or a solution of warm water and baking soda. Apply the solution with a spray bottle, wipe it with a sponge and continue to buff. A store-bought granite sealer can also be used. Follow the manufacturers instructions for how to apply.