There are many tools you can use to remove pet hair--a tape roller brush and a vacuum cleaner with an upholstery brush are two. But here is a simple method: simply use a microfiber cloth, either dry or a little damp. These cloths grab so well that they even pick up pet hair from upholstery with a minimum of effort.
Here is a classic from a classic. Rodale Press online, the venerable back to the earth organization, has published a short compendium of classic recipes for cleaning agents, everything from multi-purpose cleaners to tile cleaners to window cleaners and beyond. The recipes also include essential oils.
This article really goes into the physics and chemistry of dirt in grout--a substance that has peculiar properties that are hard to clean with just one approach. For instance there is usually soap scum settled on it so that what is stained lies underneath. Listed are good alternatives to bleach, particularly hydrogen pyroxide. http://www.mnn.com/your-home/at-home/stories/secrets-to-cleaning-tile-grout
What a great idea, right? Right. Try vinegar, castile soap and hydrogen pyroxide in an automatic sprayer or a spray bottle (well almost auto). Saves a lot of scrubbing later! http://www.mnn.com/health/healthy-spaces/stories/we-tested-it-homemade-shower-cleaner