This page provides an excellent overview of radiation types and what you are most liable to encounter. For airline flights gamma-radiation is the most probable type. (Alpha particles are stopped by paper, Beta by thin foil or similar.
You can buy commercial instruments that will detect radiation for from about $200 on up.
An off the shelf "PIN diode" acts as an adequate gamma radiation detector in many cases. Here is a PIN diode radiation detector design from Maxim. Their notes are worth reading in this context. Source
A PIN photodiode, four low-noise op amps and a comparator are used to detect individual photons of gamma radiation. The schematic, design considerations and component selection are discussed.
The circuit of Figure 1 includes a PIN photodiode that detects individual photons of gamma radiation. When a photon strikes a depletion region created by reverse bias on the photodiode, it produces a small amount of charge in proportion to the photon's energy. The resulting signal is then amplified and filtered by four amplifiers and a final comparator distinguishes between the signal and noise. The comparator output pulses high each time a gamma photon with sufficient energy strikes the photodiode.
$4000 on up products
$220 product. Russian made
$200 & $350 commercial products.
Here is a description of a DIY real world Gamma Ray detector based on a PIN diode.
Made by VK2ZAY - Australian radio-amateur. A number of other on-web designs are based on his work.
Photodiode Gamma Ray detector
4 minute You tube description good.
Here is a Hackaday article describing the same article
Related videos - same device here and here and here - finished product - 10 minute video