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.
This is good advice and you shouldn't ever mix cleaning chemicals of any type, but acidic reaction is only one type of problematic reaction with bleach.
Mixing it with ammonia-containing cleaners or rubbing alcohol will also create harmful gases, and since bleach is such a strong oxidative agent, you want to avoid mixing it with anything if you aren't 100% sure how it will go.
Honestly, given how strong it is, I wouldn't use it unless you absolutely need to. I am a biologist and have clearance to work with biohazardous materials, and in our lab we mostly use vinegar (for light cleaning of workspaces) and ethanol (as an all purpose disinfectant). Bleach is only used when neither of those will suffice.
I hope I'm not sounding like I disagree with your comment... Bleach is useful for very dirty jobs and should be in every cleaning toolbox. I just also don't want people assuming that it's safe to mix bleach with other chemicals if they discover that it isn't acidic!