What's the best way to clean stained grout on the countertop & floors?

Try a 50/50 solution of vinegar & water. If that doesn't do it, use a strong bleach solution. Either will need to soak for a short while. You should seal the grout once it's clean.

I use a vinegar and water mix and a toothbrush. Once the grout is clean be sure to use a grout sealer to prevent further staining.

A simple solution using regular household bleach and water should do the trick!

Gel soft scrub. Squeeze on, leave on for a couple hours then scrub. Wipe with wet cloth, dry for at least an hour then reseal with the clear grout sealer.

I read a tip from a woman who worked as a janitor at a University, and she used toilet bowl cleaner on the grout in a tile floor.