Both forms are correct, but we use them in different situations.
You would say 'I go to the airport' if you want to describe a regular occurrence. For example, if a driver is talking about what he does in his job, he could say:
'I go to the airport every day to pick up passengers'.
We use this form (the present simple) with adverb phrases such as every day, always, never, usually sometimes, to talk about habits.
However, if you are talking about NOW, and you want to say what you are doing at the moment, or what you intend to do very soon, you would say
'I'm going to the airport.'