You can generally go get your Florida driver’s license basically immediately. You need to make sure you have the right documents (passport, proof of SSN, savvy nomad "Letter to the Tax Collector" downloaded form portal, and your second proof of residential address).

FLHSMV says:
​Name and address changes must be updated on BOTH your driver's license/ID Card and title/registration within 30 days of changing your name or address.

Although you can obtain your driver's license later, we recommend doing it as soon as possible to reduce risk and eliminate confusion with your previous state of residency.