¿Cuál es la fecha? - What's the date?
¿Qué día es hoy? - What day is it today?
When saying or writing the date in Spanish you say the day, then the date, then the month, just as in English.
Hoy es lunes, cinco de junio. - Today is Monday 5th June.
The date is said as a cardinal number eg 5 not 5th.
el 5 de junio - 5th June
el 16 de diciembre - 16th December
Es el uno de enero. - It's 1st January.