Coral are protected worldwide under CITES II (threatened by trade- e.g. without regulation, trade has the potential for serious hard to populations). It is estimated that every day 10,000 pieces worldwide are collected to supply the home aquarium trade (9,000 of these die en route to consumer, which buy 1,000 a day). This is in addition to tremendous and generally unregulated pressure on corals by coral mining for construction materials, and habitat destruction. It is illegal to transport coral across international boundaries without proper permitting documentation from CITES, plus usually an export permit from the home country and an import permit from the destination country. I have never heard of anyone getting a permit for personal use.
If you simply want the option to collect if you see something interesting (and say- find out the ID, or more about its biology), then I recommend the following scenario, which will ultimately help you get connected with the community of coral researchers/conservationists in your area and help you learn more in the long run: 1) take a picture of the coral in its natural setting (i.e. don't break off a small piece or lift it to the surface) using an UW flash- get a shot of the whole colony, plus use a macro setting to get a close up of the polyps 2) draw a good map of the location so you can return, or tell others how to return, 3) look online and in books to identify it- there aren't that many species there, so although life forms are very diverse, it should be possible to narrow it down, 4) when you find something new, share this with local experts. Florida has lots of coral experts who can help if you run into further questions- and even better perhaps you can intern with/volunteer for one of them.
A special note about corals in Western Tropical Atlantic/Keys- coral disease is rampant there. Any injury, such as from a fin-kick or from purposefully breaking coral, offers a potential entry point for disease, which can end up killing the entire colony.
Reef Check, and similar bleaching/disease watches are a great way to use dive time to further an understanding about coral and at the same time contribute to coral reef understanding. Does anyone here know who may run these in your area?
Anyway- enjoy school and diving!