I don't think FL is different than any other NFA friendly states.
First you need to find a Class 3 dealer. You can buy a suppressor from him or mail order one and have it sent to him (adds 1-2 months more wait time). You usually pay for the suppressor up front.
He can walk you through the paperwork but I'll tell you want you need to do:
Fill out an ATF Form 4 in duplicate, get your local Police chief to sign the form for you.
Get two sets of fingerprints taken by the PO (dealer should have the correct fingerprint cards)
Get passport photos taken for your ATF Forms (2 needed)
Fill out a Citizenship form (dealer should have it or you can print it off the ATF website)
Write a check out for $200 to send in with your ATF Forms
Send everything in to the ATF, then wait and wait some more (transfers typically take ATLEAST 4 months)
When it clears, your dealer will get the paperwork mailed to him from the ATF.
You go to him and pick it up and write another check out to him for the transfer ($50 is typical)