These are all great tips - I just want to add one:
The National Weather Service offers free Skywarn classes to the public every spring. The class is designed for the average person to be able to recognize (and hopefully report) severe weather and tornadoes. I feel pretty certain, that even devoid of all electronics, that I could tell when to take shelter because something ugly is about to happen; and if I DO have my electronic stuff available, I know how to recognize a tornado (or potential tornado) on radar.
To find a class near you, go to your local NWS page and look up "Skywarn". (Chances are that most areas have already held their training classes for this year)
Here's a link to the Nat'l Skywarn page
Skywarn