What chemicals and tools do I need to clean my bathroom?

Bleach is awesome! Mix it with some water, use a rag or sponge and scrub. DO NOT mix anything else with the bleach. Bleach is basic, and a lot of other cleaning products are acidic. Mix a strong enough acid with a strong enough base and you get mustard gas. It doesn't smell too great, on top of other issues. If you want to clean with bleach plus something, be safe and buy premade cleaner with bleach.

EDIT: I should also say you can't use bleach on everything, it can damage some things. Always test a small, hard to see area before you clean a surface with bleach. For the stuff that you can't get bleach on, you can use white vinegar or all-purpose cleaner.