"gonna" = "going to" + [verb] : indicates some action in the future. "I am going to [gonna] drive to Beijing tomorrow."
"gotta" = "have got to" : must or "have to". I have to [gotta] leave now.
"wanna" = "want to"
What Norman says is partially true - these words are used in spoken English, not written. However, in spoken English, I hear them used a great deal - even by people who normally speak English well. You would not hear them used in any formal setting, such as a public speech or meeting. In normal conversation, though, they are used quite a bit. I would eliminate them from your written English.