Neither. They both look horrible.
Gonna is a feature of spoken English. It's just a sequence of sounds heard in spoken English - it is not part of written English. Nobody will take you seriously if you write English in this way.
The question that you are looking for is this:
"What are you going to do today?".
Neither is correct in WRITTEN English.
In WRITTEN English, you must write:
"What are you going to do today?"
In SPOKEN English, you can say:
"What are you gonna do today?"
and that would be correct.
However, if you say:
"What you gonna do today?"
this is bad grammar. If somebody speaks like this, they probably do not have a high level of education.
Sara wrote this: 'Gonna' is slang for 'going to', so that means that in formal conversations, form 'gonna' is not correct. You can use 'gonna' in informal conversations only!
Sorry, but this is not correct. Are you suggesting that we English speakers pick the word 'gonna' out of our mental vocabulary books especially for speaking to our friends and family? Or that we replace this with 'going to' to show that we're being polite when speak to our bosses or lawyers? And that this is reflected in our written language? I'm afraid to tell you that this is nonsense.This issue has nothing to do with slang, and it has nothing to do with informality. It is about the difference between the spoken and the written language. Most of us say the sounds represented by gonna - or more commonly a sound that's more like g'nuh - when we're talking, but we do not write this. Song-writers, cartoonists and caption-writers write gonna to show us they are representing spoken language, but ordinary English speakers don't. If you don't believe us, look through a few posts on italki. Who writes gonna and wanna? Only non-native English learners who've been misled into thinking that this just informal English. And what do native English speakers write? We write normal English - going to and want to.