Summarize:

Before getting down and scrubbing your grout, you need to do a basic once-over with your favorite cleaner. If you cleaning floor grout, sweep and mop. For bathroom and kitchen counter grout, spray, and wipe it down with your favorite cleaner. In a small bowl, mix hydrogen peroxide and baking soda to form a thick paste. The parts of each may vary depending on the consistency and thickness you prefer to work with. Use your finger or a toothbrush to spread the paste onto your grout. Start in a small area only, no larger than 1-2 square feet. Keep the solution thick and completely the coat the grout. Allow it to set for 5-10 minutes. Use a small brush such as a toothbrush (electric preferably) for scrubbing at the grout. Apply firm pressure over a small area to lift dirt and stains. If the grout is still dirty, add more of your paste and scrub again after letting it set for a few minutes. Continue adding the cleaning mixture to the grout and scrubbing it off. Work slowly to ensure that you have gotten all your grout cleaned up. Use a damp rag to wipe off the remaining paste on your tile. Finish by cleaning your tiles as usual with a spray cleaner for counters or a mop and soap for floors.
Clean up your tiles. Create your paste. Spread your mixture. Begin scrubbing. Finish your space. Wipe up your tile.