How do I get the current date and current time only respectively in Django?

For the date, you can use datetime.date.today() or datetime.datetime.now().date().

For the time, you can use datetime.datetime.now().time().

However, why have separate fields for these in the first place? Why not use a single DateTimeField?

You can always define helper functions on the model that return the .date() or .time() later if you only want one or the other.