Great question, Alberto. I agree that you should always use the official spellings ("want to," "got to," and "going to") for these common expressions. That being said, in the spoken language, "wanna," "gotta," and "gonna" are absolutely the usual pronunciations (in all dialects of native English -- they are not Americanisms -- they are the inevitable result of English phonetic rules), and I highly recommend that you familiarize yourself with them so that you will understand standard spoken English. I strongly disagree with English teachers who insist on clinging to silly notions of prescriptive linguistics and thus fail to prepare their students for real live conversations. If you follow their advice you will be well prepared to understand English teachers, but not a normal conversation.
You should also learn: "Hafta" and the reduced form of "can" (/kn/).
After you've learned the past unreal conditional, you should also familiarize yourself with "coulda," shoulda," "woulda" (note that the "l" is silent in each of the preceding three words), "mighta," and "musta." Plus the negative forms, but already we're getting into rather advanced grammar.