We get so many Tornado watches and warnings around here that it doesn't faze me much. I will pay attention to warnings, taking note of locations, directions, and speed. If it's headed my way, I pay VERY close attention to it. If it's not headed my way, I go back to what I was doing.
However, a twister "out of the blue" is indeed a troublesome event, no matter how big or small.
What they teach us around here is to take shelter in an interior room on the ground floor, such as a bathroom or coat closet. If you are outside and one spins up from "out of the blue", take shelter in the lowest possible spot you can, such as a ditch. Underneath overpasses are NOT a good idea, as they create a wind tunnel effect.
Around here, they've started selling "safe rooms" that can either be built into a new home, or added to an existing home, typically in the garage. They bolt down to the concrete pad / foundation, and the walls are strong enough to resist impact damage from EF-5 winds. This, or course, is in addition to storm cellars (commonly called "Fraidy Holes") that were dropped into a lot of the back yards of older homes many years ago.
I have a "disaster kit" in the master bathroom (our "fraidy hole"). It consists of a battery-powered radio, blanket, and a few candles. I made that many, many years ago, right after the May 3td 1999 outbreak. This is the system that spawned an F-5 that went through Moore, OK, and OkC OK. The last tornado of the day came within 1/4 mile of my place at that time.
I really need to update that kit. I believe I used all of the candles when we lost power last December in the ice storm. We do have a few more battery-powered flashlights and lanterns. My radio was actually one of those hand-cranked rechargables, AM/FM, flashlight, strobe, etc. The dynamo on it is broken. I'll need to see about replacing it. I really should get some cheap shoes that everyone can wear, in case we have to take shelter before we can put on shoes. A case of bottled water is always a good idea, as are a few MRE's.