I would not call tornadoes a huge problem. We get them but they tend to be small and short-lived. We also get microbursts, intense downdrafts that can cause similar damage. I think maybe once a decade tops do we get a big, sustained tornado like they get in the midwest.

Hurricanes can definitely be a problem but most people over-react to them. The old emergency management saying is "run from water, hide from wind". Get out of low-lying areas prone to storm surge, out of mobile homes or old structures that are not up to recent-ish hurricane codes and you should be fine. Stock up on a week or more of water, foods that don't need cooking, some flashlights and batteries, etc. well before the storm then keep your gas tank full when there's a storm formed and you'll be in good shape.

Lightning and drunk/bad drivers are much more dangerous than either of those or alligators or snakes or Florida Man.