What's The Best Way To Remove Old Grout From Between Bathroom Tiles In The Shower Area?

You can purchase a small tool from your local hardware store specifically made for this purpose--they are very inexpensive, maybe $3-5 USD.  Use the tool in several passes over the grout until you have exposed the part of the edge of tile that is not glazed.  It is carbide tipped, and sharp, so be careful that you do not accidentally slip and scratch your tiles.  An alternative to these tools, you can use a small slot-tipped screwdriver and a small hammer, but it is not as efficient and is more difficult as well.  Be careful of the pressure you apply if you choose this way.

Removal of grout primarily needs a firm brush. Apply some amount of bleach with the brush that is just enough to get thoroughly wet and allow it to settle in for a while and scrub a little with the brush. The bleach will leave your tiles and whole bathroom sparkling. Good luck.