I am new to Florida too - moved south a few years ago from the northeast.
I see Bougainvillea growing everywhere. It is beautiful but prickly.
I have a porthos( I think) from a friend that grows like gangbusters. I grow mine as a hanging plant. I have to keep trimming it before it reaches other plants and takes root. It would climb but would also take over the area. I have seen similar plants climbing trees in untended areas.
I love roses. I would as for rose suggestions in the rose forum. I have a couple potted roses, no climbers (yet). I am still learning to keep plants alive in Florida.