When removing old grout by hand, one method is to use a carbide-tipped scoring tool. Choose the size appropriate for the spacing between your tiles. Smaller is better as the carbide tip easily scratches the surface of glazed tile. Drag the tip of the scraper down the edges of the grout, cutting it out from between the tiles. Finish up problem areas with a small flat screwdriver or a wood chisel and a hammer, gently chipping out any remaining grout, angling the chisel from the center of the grout joint into the tile to avoid chipping the edges of the tile.
For large jobs, a more expensive option is to use a grout saw. These require some expertise to use effectively, safely, and without damaging the surrounding tile, but in the right hands, they are ideal for large jobs. Even when using saws, you may still require the hand tools to clean up edges, crumbled areas or problem spots.
If you have an area where the grout is still holding up but has become discolored, use a commercially available grout tint to paint the grout. Although laborious, the results are often well worth your efforts, especially in small areas. Use a fine artist's paintbrush to apply the color, being careful not to get it on the surrounding tile.
Protect the tile itself if it is on the floor, as well as the tub or shower basin, by throwing down a tarp or a sheet before beginning a grout removal project. The sharp crumbles easily damage any surface if ground into it. Also wear protective eye goggles when chipping out old grout as wayward pieces may unexpectedly become projectiles.