'Today I do my homework' is natural only of you see this as part of a schedule:
Here is my plan for the week:
Today, I do my homework;
Tomorrow, I revise for the exam;
Wednesday, I tidy my bedroom.
If you are referring to a plan or arrangement, the natural things to say are:
I'm doing/going to do my homework this evening.
If it's an instant decision, or promise or offer, it's more likely to be;
I'll do my homework this evening.
You might find this interesting if you are discussing such matters with your colleague.