How can I seal the grout on kitchen wall tiles to give lasting stain and mold protection?
You want to apply a penetrating grout sealer. This one says that it last up to 5 years depending on conditions.



The key for the sealer to last as long as possible on a backsplash is to avoid cleaning it with "harsh cleaners" like bleach and antibacterial sprays.  You want to use something that is not going to remove the grout sealer when cleaning the backsplash.  They sell specific cleaners that are easy on the grout at the tile shops, but I have read that just using diluted white vinegar works.