import datetime
variable = datetime.date.today()
This works as expected.
from datetime import datetime
variable = datetime.date.today()
This gives the error you described. That's because datetime.datetime.date is a method, not a class. datetime.date.today is a method, but not datetime.datetime.date.today (the extra datetime is the problem).
First thing you should do is launch an interpreter (type python at a command prompt) and type just the first two lines. It should work. (if it doesn't, I would probably just reinstall python)
If it DOES work, the problem is in your code file. You must have something overshadowing one of those classes. Since you didn't provide the error message from the first code example (which is correct) I have no guesses as to which class it is... but carefully look at your import statements and variable definitions between import datetime and when you try to use it.