A cuanto estamos hoy? < I don't know if this phrase is correct or not but it is very common, in english it means -At how many (days) are we today-So the answer is something like "Estamos a 15" -We are at 15th- In this question is already implied that you know the month and year.
Cual es la fecha de hoy? =What date is today? this one is very clear
Que dia es hoy?=What day is today? < this is asked to know the day of the week.
Now in adition to the ones above A que estamos hoy? Is also very common and in this one you need to say the complete date .It means -At what (date) are we today?