There isn't really an 'FBO' in the usual sense ie the FBO's own building with all amenities incl. crew area & kitchen/galley. Unless Odyssey have expanded to Freeport too. In which case ignore most of the following...
Park on the apron with the other GA aircraft near the demountable buildings just east of the terminal. Often there'll be someone to help with bags, but not always (don't forget to tip. A tip seems to be expected by everyone for everything). Just inside the door is Immigration, Customs & Airport fee agents. Pay *each* one their fee. It used to be that you could only pay Immigration & Customs each in cash but use a CC for the airport fee. It's been a few years since I flew there regularly, and not recently, but I seem to recall that they've changed the system a bit and now you can pay it all in one hit by CC to an FBO-ish arrangement at a desk in the pax waiting area on the other side of demountable.
Weather briefing is in the same building in a room opposite customs & immigration. A phone is there to call US customs & immigration to notify arrival.
Paperwork. You'll need:
Arrival report, Gen. Dec x 3, customs/immigration card for each person & a pax manifest. If you're going to be flying within the Bahamas you will also need a 'transire' completed, the aviation equivalent of a cruising permit.
All the forms - except maybe the customs/immigration card - are available on the Bahamian government's website. Don't have the URL but Google will find it.
If you're exiting or entering the USA then don't forget you'll need an eAPIS account and **MUST** submit details via their website at least 60 mins prior to departure from the US or arrival at the 12nm ADIZ. Strangely, the supposedly wonderful eAPIS notification system still has individual airport C&I facilities in the US needing a phone call at least 60 mins prior to arrival. A phone for that is available in the demountable.
When leaving you'll need 3 x completed Gen Decs. They'll give you 1 x stamped copy back which will show your departure time. Don't dawdle too long aftwards or US customs might wonder why your flight took so long...
You will also have to submit a Bahamian flight plan via either the airport payments desk or the FBO desk, depending on how they've organised things now. Don't forget to submit a US flight plan in the normal way eg using DUAT, DUATS, 1-800-WXBRIEF etc. I use
FltPlan.com Flight Planning & Flight Tracking. for all my US flights.