Are you going today? or Are you going now? is the best choice.
Do you go today? is not often used in American English and can sound a bit awkward. This is because a typical response would be, "Yes, I am going today," using a conjugation of 'to be,' not 'to do.'
However, other verbs can use 'to do' and 'to be.'
Are you leaving today? and Do you leave today? are both correct.
BUT, if the time is now and not today , use Are you... . It will be less awkward.
Are you leaving now? is a better choice than Do you leave now?