They are not standard English, and are written only within quotation marks to show the precise way a person is speaking. Recognize them, understand them, don't try to make them part of your own active vocabulary. They are a shortened, blurred version of "going to" and "[have] got to," the natural result of speaking quickly. "I'm going to" means "about to do something in the future." Correct English: "I'm going to get a cup of coffee, do you want to come with me?" Colloquial spoken English: "I'm gonna getta cuppa coffee, wanna come with me?" "[have] got to" means "need to" or "should" or "ought to," an obligation. Correct English: "I need to get something to eat, I'll buy a sandwich or something else from the vending machine." Colloquial English: "I gotta get sump'n t'eat, I'm gonna hit the vending machine for a sandwich or sump'n."

gonna = going to gotta = Got to (or 'have to') Very, very casual spoken English. One would probably only see it written where an author was trying to impart how someone sounded when they were speaking.