Most of the SkyWarn activities are in the spring, with training generally late winter or early spring. If you back up one page from the link you provided (
NWS Jacksonville, FL [Skywarnâ), they list some training classes, which may be the best way to start out.
I notice that your link is for the JAX NWS office. There's also one for Tallahassee at
TAE Spotter Information which may be easier for you to get to. They appear to have an on-line class on 11/8 that may also be a good place to start out. You may also try contacting one of the ham clubs in your area or check in to a local ARES or SkyWarn net if they have one.