E.A.
Old stains from dry erase markers on a white board? Dampen a rag with rubbing alcohol and it will come right off. Is it really stubborn? Use hand sanitizer, let it sit for 10 minutes and try again with a plastic scrubbie. The hand sanitizer won't run all over the place and won't evaporate as quickly.
I try to clean the bathroom right after someone takes a shower, it's always easier to wipe the mess up.
Don't clean up cat puke off the carpet until it has dried, it will crumble and you can just pick it up with tissues, then vacuum up the last of it. Trying to clean it up when it's wet just smears it.