Its my understanding that state-run testing sites are free to all as part of the CARES Act. If its a non-state run site (i.e. Walgreens, urgent care, etc), then many insurance companies will only cover the cost if you are symptomatic and/or referred by your doctor.
I looked into this because we are not symptomatic and were hoping to do the test to end our quarantine early, but there were no state run testing sites near us so we would have had to pay upwards of $300 for it. But, it doesn't matter because its effective beginning Wednesday so we are not eligible.
As PP said, its both. You have to obtain a test within 3 days of leaving FL. Then quarantine in NY for 3 days when you get home. Then go get another test in NY on day 4. If both the test from FL and the test in NY are negative, you are done with quarantine.
I'm waiting to see more details from NY, but in MA, the paperwork says the test has to be taken within 3 days of leaving the state you're in (FL, for example) but you're not required to have the results in hand when you board the plane. The paperwork you are required to fill out upon returning home has you check a box that you completed the test, but do not have the results. The health department then follows up with you to obtain the results. Presumably, NY will have a similar approach and will just need you to produce two negative test results (one from FL and one from NY) to support the end of your quarantine. With testing sites taking 3-5 days to provide results, they couldn't require you to take a test within 3 days of leaving and then expect you to change travel arrangements if you don't have the results yet. It would make this change impractical for people and would be difficult for people to do.