Fit a grout supercutting tool with a diamond tipped blade. Put on protective goggles and a ventilation mask.
Turn on the supercutting tool and insert the blade into the first vertical grout line. Slowly pull the tool down the wall, digging out the grout to a depth of at least 1/8 inch. When you cut out the vertical grout line, switch to cutting out the uppermost horizontal line so your arms do not get as tired.
Repeat this digging out of the grout lines in the shower, alternating vertical and horizontal lines, until you remove all the grout lines. If there are any lines you cannot reach with the supercutting tool, dig those out with a regular grout saw with a steel blade.
Go over all the grout lines with a stiff-bristled brush to remove any loose grout and dust. Run the hose of a shop vacuum over the grout lines.
Mix your grout in a bucket with water according to the grout package directions. Use a grout float to scoop grout out of the bucket and deposit it onto the shower wall, starting at the top of the wall in a corner. Hold the grout float at a 45 degree angle to the wall and drag it diagonally across the tile surface, forcing the grout into the lines. Grout as much of the tile as you can in 5 minutes.
Dip a sponge into a bucket of clean water and run it over the surface of the grouted tile to remove any excess grout. Rinse the sponge and repeat the washing of the tile surface, if necessary.
Continue grouting across the surface of the tile shower horizontally, then move down closer to the floor. Only grout as much as you can in five minutes before pausing to rinse the tile surfaces. Change the water in the rinse bucket when it becomes cloudy.
Let the grout cure according to the directions on the package.