The relationship between "get" and "have" is so obvious to a native speaker that we may have trouble actually seeing it clearly. I don't know technical terms to explain it, but I would put it thus...
"To have" is essentially a verb describing state, like "to be". "To get" is a verb describing action, like "to become". Indeed, these may be considered two verbs that provide examples of the same relationship. If you are not rich, you may become rich, and then you are rich. If you do not have something, you may get it, and then you will have it.
You may come across a rather hackneyed phrase, where someone (usually a man) is so fixatedly attracted by someone else (usually a woman), he says "I must have her!" This indicates that he wishes to be in the state of possessing the woman (an awful way of thinking about relationships, but there you go). In order to enter that state, he has to "get her", and thus a foil in some work of fiction might then ask "well, what are you going to do to get her?"
People in real life rarely talk like that, though they might think like that.
Now, dialectal usage has the expression "get with". To get with someone is to become romantically involved in some way - the first example that comes to mind is the song No Scrubs by TLC, back in the 90s - "Wanna get with me with no money / Oh no, I don't want no scrub". I've not personally come across a similar usage that elides (misses out as implied) the "with", but it wouldn't surprise me. On the other hand, it could simply be reflecting a possessive attitude.
In any case, whether it reflects the idea of possession or the dialect "get with" expression, "get" represents a progressive action or a change in state, while "have" represents a steady state.