wanna = want to gonna = going to The difference is the level of formality. "wanna" and "gonna" are fine to use with friends, especially in fast speech. They are also ok in some online chats or text messages, but they are mostly used among teenagers and young adults. These words are never ok in academic, business, or formal writing.

wanna = want to I want to eat. I wanna eat. gonna = going to I'm going to eat. I'm gonna eat. Still same meaning, but wanna and gonna are more for casual conversations.

Although some people might tell you not to use these expressions because they are slang or lazy, I believe that if you really want to learn a language, you should aim to be as near a native speaker as possible. That is the real language. I'm a native English speaker, and use both expressions, although "gonna" sounds more like "gunna" in normal speech. I'm also a fluent (Mexican) Spanish speaker, and use expressions like "pos", instead of "púes", and "dónd' estás", just like native Mexicans. It's just a normal part of TRULY mastering the language and all its idiosyncrasies. Real language doesn't come out of a textbook!

There's one thing that people haven't mentioned. gonna = going to, BUT ONLY IN THE STRUCTURE BE GOING TO + VERB DON'T say "gonna" when you mean going to + place. "I'm gonna go to a party tonight" = OK (going to + go) "We're not gonna make our flight!" = OK (going to + make) "I'm gonna Hawaii for winter vacation" = WRONG

If you are going to learn English, you might as well learn to speak correctly, which would be appropriate for any conversation, casual or formal. I don't see why you would want to learn an incorrect version, or slang version, especially if you want to learn to spell and speak well for all occasions. Say "going to", which means you intend to do something, and "want to" which means you would like to do something.