You can buy a commercial off-the shelf "neutron generator", or you can use a radioactive source.
Neutron generators are accelerator-based fusion reactors1 and have the advantage of being able to simply turn the neutron supply on and off.
The most common source is AmBe (Americium-241/Beryllium), though Californium-252 and tritium both have their uses. I also built a Po-210/C-13 source once as part of my work on KamLAND. Sources require no external power supply and no maintenance (though they do decay away if you use a short half-life progenitor like Po-210), but they run continually.
You basically can't "catch" free neutrons in the sense of collecting a bunch of them in one place.
On the other hand, thermal neutrons (those whose kinetic energy in on scale of that of room temperature gasses) can "capture" on a variety of atoms. Hydrogen-1 and Carbon-12 are the most common nuclei that can capture a free neutron. Chlorine, boron and especially Gadolinium are useful because they have high thermal neutron capture cross-sections.
1 Yes, fusion reactors. They've been around for many decades. The fusion "problem" is in getting ones that generate more power than you put in.