Well, you know that the classic Tornado Alley starts up in Texas in mid-April and the activity moves north-northeastward through early summer and Dixie Alley usually runs March through April. Although of course tornados can occur anywhere anytime the frequency is usually seasonal so the obvious thing to do is avoid those areas when appropriate. This is an anomalous year so far.
https://en.wikipedia.org/wiki/Tornado_Alley#/media/File:Tornado_Alley_Diagram.svg
https://en.wikipedia.org/wiki/Dixie_Alley#/media/File
ixieAlleymap.png
As for Florida (and including coastal Georgia and the Carolinas) we don't get those long-track EF-5 monsters that routinely eat cities in Mississippi and Alabama; there's really not a whole lot of likelihood of getting caught in one.
So I'm in east-central Texas right now and I won't head northward and eastward until around June 1; I'd rather let the tornado seasons wind down than worry about it.
Strong straight-line winds can occur in any big thunderstorm anywhere, as can hail, but big hail is fairly seasonal with the tornado seasons.
Hurricanes are of course different; you'll have plenty of warning and the thing to do is drive inland maybe 100 miles.
You should get a weather radio that can be programmed to alert to the SAME settings http://www.nws.noaa.gov/nwr/coverage/station_listing.html so you can set it up when you're in the same place for a while. When traveling you'll need to find a station and be aware where you are. Also I believe Weather Underground can push alerts to smartphones. On the computer there's forecast.weather.gov and wunderground.com .
When you get a CG find the storm shelter (if any) or a solid building so you'll know where to go if heavy weather is coming your way.