I've never used a film camera before and don't know much about cameras in general so I wouldn't know how to check or anything.
Before doing anything else, I'd recommend reading the manual to familiarize yourself with the operation of the camera. (Note that there's nothing automatic about this camera! You need to meter using an external meter or sunny 16, set the shutter and aperture manually, and wind manually after each shot.)
The Canon III is a beautiful old camera with a simple design. There are basically three things that can break: the shutter, the winding mechanism, and the rangefinder. You can try the following for a crude test.
First, check that the shutter and winding mechanism work. Take the lens off the camera. Without any film in the camera, cock the shutter by turning the winding knob as far as it goes, select a shutter speed, press the shutter button and look through the lens mount. You should see two curtains moving. Repeat, trying all the shutter speeds. With lower shutter speeds, there should be a perceptible lag between the movement of the two shutter curtains. With the 'B' (bulb) setting, the curtains should stay open as long as you keep pressing the shutter button. If any of this fails (e.g. if the curtains always open for the same duration regardless of which speed's selected, or if the camera jams), there's likely to be a major mechanical problem.
Next, put the lens back on the camera. Looking through the viewfinder, focus the lens at different distances and observe the rangefinder patch. You should see two images moving relative to each other (when they're overlapping, the object is in focus).
If these tests check out, the odds are that everything is in order. Load some film and enjoy your camera! (Note that early rangefinder designs like the Canon III can be a bit finicky about accepting modern film. See this question for more advice.)
Finally, note that your camera is about 70 years old. While these things were built to last, they do require some maintenance every now and then, depending on use history. It is quite likely that, even if your camera passes the above tests and is capable of exposing film, the fine adjustments are out of calibration by now: the shutter speeds may be erratic and the rangefinder may be out of adjustment. If this is the case, you may want to consider taking the camera to a technician for a CLA (clean, lube and adjustment).