There are different styles of grout scrubbing tools, but they all involve some kind of handle connected to a very strong metal piece, like a wheel or triangle. Using it is pretty easy -- you simply run the metal piece between the tiles until you remove the outermost layer of grout and scum. Follow behind with a wet sponge and be prepared to use a little elbow grease. It's going to take you a while, especially if you have a large bathroom, but the result is worth it.
Because it's not a very fun job, the best tips for grout scrubbing and cleaning involve not letting the grout get that bad to begin with. Spot cleaning can be easily done with a baking soda and water paste applied with a toothbrush, or even the eraser of your No. 2 pencil. But no amount of cleaning beats sealing a new grout job. This protective layer will keep moisture out, which creates the mildew that darkens the grout. If you're really diligent and apply the sealant every six months or so, you may never have to use a grout scrubbing tool at all. Another tip is to think about going with a darker tile, which supports a darker grout. Or, if you're set on lighter tile, consider using a light gray grout instead of white. White grout requires a pretty serious commitment to regular cleaning and sealing.