It's truly sad that these days you have you yell and get upset before anyone takes you seriously.
Maybe at the end (or beginning, just as long as it's really clear) of any conversation regarding an issue you have, say something like "This is the first/second/third time I'm bringing this to your attention."
Perhaps when the contract is written/relationship begun, you can explain that when you're unhappy with a situation you will bring it to their attention calmly and rationally and that shouldn't be an indication of the level of seriousness - you just choose to behave professionally in a professional relationship and not yell or get visibly upset.
Update: you've added information since I posted - three weeks may not *typically* be enough time to see a change in a classroom situation. You added at one point that the teacher was basically completely awful and his contract wasn't renewed - meaning things probably wouldn't have ever gotten better. However, the director may have been basing her timeline on a reasonable adjustment period for a teacher to make necessary changes.
As for the nursing home - you say "so he better get out of my way." That seems an odd statement...
As another poster said, we're only getting your side of the story. You sound reasonable - but are your timelines/expectations reasonable? Do you clearly lay out what you expect to happen? You said you told the yard guy's employees some of the issues - you can't really be sure those issues were brought back to the big guy in charge.
Obviously, based on the information for the yard guy, you were completely reasonable in expecting him to live up to his end of the contract and do what you were paying him to do. He sounds like the type who tries to manipulate situations and people as Angela S. said.
If all else fails, simply document in writing every time you have an issue or complaint. Most business take written complaints seriously, regardless of tone.