Notice the discolored grout in the white grout lines, most of what you see is easily taken care of, it just takes patience and perseverance and it can look like new.
There are a few different sizes of grout saws you can use, the one above is very inexpensive but fits comfortably in your hand. The one below is bigger but essentially does the same job. You can purchase these at any homeimprovement or your local hardware store.
The object here is to run the grout saw back and forth the grout between the ceramic tile, at differen intervals angle the saw to each side as to get close to the tile without scratching it. Do not apply to much pressure, all you want to do is scratch the surface to remove the soap scum and or mold.
The tedious part is going nack and forth and up and down in every groove or grout line. The best thing to do is do all the verticals then all the horizontals on a wall or you can break it up into sections.
You can always take a break from running the saw and use the single edge razor knife over the individual ceramic tile to also remove the soap scum from the surface of the tile.
Once you get an area done, just wash it down with a soapy scrubby and rinse clean. You will be amazed how good the grout will look.