Help selecting component to detect red laser light
I assume the 'training pistol' is hand held. 

If it is, you likely only need detection in several milliseconds, or more. 

Almost all photodiodes and phototransistors are fast enough by a factor of 100x, or more. So, as Scott Seidman says, fast detection is not likely an issue. 

I'd recommend starting with a visible light phototransistor. They are easy to use as detectors, and are cheap, i.e. under $1. (e.g. Vishay TEPT4400). It is over 90% sensitive at about 630nm. These have a wide enough range of wavelengths that they will still pick up infrared light, so use an optical filter, to reduce the intensity of other light sources too.

To avoid confusing the detection of the laser with sunlight, or some other light source, the laser needs to be modulated. This needs to be done in such a way that it is easy to discriminate the laser signal from other light sources.

Simple hand-held laser's can be modulated at about 5kHz, which is well away from mains-powered light modulation (50, 60, 100 or 120Hz), and Infrared remote control (36kHz or higher). Depending on your level of skill, you might build the filter using 'pure electronics', or you might use a small microcontroller.