No matter how clean you keep your bathroom, the grout in between the tiles will still become discoloured and worn at some point. Although some discolouration can be treated, when grout becomes heavily worn and cracked, it is time to replace it. Removing old grout is easy with the right equipment and some preparation.
Use a grout saw to saw in the spaces between the tiles. Grout saws are designed to work in the narrow spaces between tiles, and are available at most hardware stores.
Use a chisel to clear away remaining grout. You may need to use a utility knife to cut closer to the tile if you are unable to remove all of the grout with the chisel.
Sweep up stray chunks of grout and grout dust with a broom. Go over the area with a vacuum hose to clear away any remaining debris.
Once the area is cleaned, inspect the tiles for chips or cracks. Replace any broken tiles before replacing the grout. If the area is excessively dirty even after sweeping and vacuuming, you may want to go over it with a light detergent (such as a dish washing or washing powder) and a clean, wet rag.