To write the word 'gonna' is extremely informal. For some reason Chinese people seem to think that it is a lot more common and acceptable than it really is. We may say 'going to' as 'gonna', but we don't generally write it like that. In your example, it is even more out of place, as the other language is in a slightly formal register. You could write 'I am going to go', which is a slightly different meaning, but is an appropriate way to write it.

I agree with previous comments generally. Need to go is to have reason to go. Gonna go means you are going to go (even if you want to just look around without any real purpose) Also note "gonna" is generally regarded as an Americanism and is not used much if at all in the UK and may even be frowned upon as many Americanisms are! "Have a nice day" (being another good example)

You could say "I gotta go..." for a more informal version of "I need to go"

They have completely different meanings: "need to" means that the action being done is necessary or required. "gonna" is simply a colloquial or slang form of "going to" which simply indicates an action done in the future.