IMHO there is a lot to be said to possession of a Geiger counter, and preferably more than one.
More than one is desirable for a second opinion so as to avoid a false sense of security, or indeed a needless panic.
Most types are not terribly accurate, but that is not the idea, it is to give warning of levels significantly in excess of normal.
There is a lot that can be done if an abnormally high count is detected, even without any elaborate preps.
Firstly try and find out from media reports and your own observations if the problem is localised or regional.
If you sea or hear an explosion and the count rate trebles or worse, that suggests a terrorist attack or some form of fairly serious accident. Get someplace else URGENTLY. Even a few miles should help for a localised incident. In an urban area you might be able to simply take a bus or taxi away from danger if you act promptly before a few thousand others have the same idea.
For a regional event that can not be realistically escaped from, then shelter in place. Whilst construction standards vary a lot, as a VERY ROUGH GUIDE it may be said that being indoors in an inner room of a normal home reduces the dose to about one tenth of that received outside.
A domestic type of semi basement, or the inner parts of a modern office block reduce the dose to about one hundredth of that in the open.
A deep concrete basement of substantial construction, but NOT incorporating any specific radiation protective measures may reduce the dose to one thousandth of that received outside.
A purpose made shelter would give still better protection.
I have a couple of small and relatively cheap Geiger counters, "GEM RADALERT II" these are simple to use and give a simple bargraph display of coloured LEDs.
GREEN=NORMAL
YELLOW-PREPARE TO PANIC
RED=PANIC.
I purchased these new about 10? years ago. At the time these were issued to some members of the emergency services, and were selected for being simple and robust. The only control setting is a choice of averaging over 10 seconds, quicker if immediate danger is suspected, or 10 minutes which is much more accurate but you have to wait 10 minutes before getting a result.
These use a standard 9 volt battery and fit in a shirt pocket.
They detect the radiation from tritium lights if placed very close.
I also have a couple of larger and more sophisticated Geiger counters with a digital display, that can show dose rate per hour, or total dose per day/work shift, or simple number of counts. These operate silently OR with an audible "beep" for each count, and also have a programmable alarm that sounds when the dose rate or the total dose exceeds the programmed limit. These instruments require some knowledge of radiation physics to get best use. They also use standard 9 volt batteries and fit in a large coat or overall pocket.
http://www.perspectiveinstruments.co.uk/ These are ones that I have. The larger model is now discontinued.
but by the time you would pick up meaningful radiation readings, you would probably already be in trouble. Unless you are near 'sources' quite a lot? 
