"I got" means that you acquired something or came into possession. You can say "I got a new phone."
"I have" means that you already possess something. For example, "I have a good job."
You cannot use "got" alone to mean that you already own something. (Saying "I got a good job" means that you were just given the job. I does not mean that it is something you have.)
It is also possible to use "have got" in place of "have." "I have got a good job."
From what I understand, this is more common in the UK than in the U.S. In the U.S., you will hear the contraction ("I've got") frequently, but not the phrase "have got," unless someone is using the phrase to mean "have to." (That's another use for it... confusing!)
Please let me know if you have any more questions!