In addition to the issues regarding proper nouns mentioned in this question and its answers, there are some usage notes specific to the words you've asked about.
I'm going home.
The speaker is going to their home.
I'm going to a home.
"A home" would be usually be interpreted as some sort of assisted living facility. A person with mental illness or an elderly individual may be put "in a home".
I'm going to the home.
I would not know what to make of this sentence. The home? Which home? What category of homes are we talking about that I would automatically know to which one you're referring? You mean your home? The home you put your grandfather in? What?
I'm going to school.
If the speaker appears to be leaving and heading elsewhere, then I would conclude that they are on their way to the school that they attend.
If the speaker is staying put, then I would take this to mean that they are enrolled in classes at a school.
I'm going to a school.
I would probably take this to mean that the speaker is enrolled in classes, but would find the phrasing a bit odd. I would likely ask them which school.
I'm going to the school.
Similar to 'home'. Which school? What are you talking about?
Perhaps this would make sense if there was a particular school that was obvious from context. You might also say this if you were going to a school building for a reason other than going to classes, in order to emphasize that distinction.
I'm going to airport.
This is definitely wrong, unless the speaker was actually referring to some place named 'Airport', like a bar or club.
My first instinct would be to ask if that was a street name.
I'm going to an airport.
This would be an odd thing to say, but would be understandable. It's odd, because one rarely wants to go to just any airport. It's not like a supermarket where you can find pretty the same stuff whichever one you go to. You need to go to the airport which has the flight for which you have tickets.
Maybe someone might say this if going to any airport fulfilled some interesting criteria, like maybe the first time they had been to any airport, or some fortune teller told them they'd meet their future spouse in an airport.
I'm going to the airport.
This is normally what people say. This is because there is usually only one nearby airport, which almost everyone in the region uses for all of their air travel.
If there were multiple airports nearby that one might potentially use, then you would use the airport's actual name instead.