Then what do you do with the shadow reading? Do you set the camera to match the meter reading, or do you stop down by some amount? And is your meter set to the film ISO or something else?
The whole idea of basing your exposure on a shadow reading is to "place" the shadow where you want it by choosing the exposure that will render the desired detail and tonal relationships in that area. This can vary from situation to situation.
The classic Zone System placement is for a "detailed black" (think a black cat or suit in which you can still see contours, wrinkles, hairs, etc.). That placement is usually two stops underexposed from the meter reading. This placement is a good guideline and starting point.
However, as we mature in visualization, we often want our important shadow area (the one we're basing exposure on) to look different. Sometimes we want it a more featureless black, in which case we place it in Zone II (three stops under from the meter reading) or even Zone I (four stops under). Or, we want a more luminous shadow, in which case we place it in Zone IV (one stop under the meter reading). For shadows on white snow, I'll often place the shadow in Zone V.
All this, of course, depends on visualizing how you want the final result to look and knowing what to do to get what you want (i.e., having tested to find your ideal E.I. and development time(s), etc.
As for metering itself: most of us Zone System users have a meter with a handy sticker added that allows one to simply place the EV value from your meter directly opposite the Zone you want the value placed in. That means no figuring how many stops to underexpose from the meter reading (not that this latter is that difficult

). No need to fiddle with exposure compensation or changing ISO from shot to shot.
All of the above works best for sheet film and using a hand-held spot meter. That's what I do. Using sheet film also allows one to choose different development schemes depending on the contrast desired. This isn't very practical with roll film. With roll film it's better to find a "normal" (N) development that works for everything. Contrast in the print is then controlled with paper grade / VC paper or in post.
When I get a small-format camera with a built-in meter in hand, I meter entirely differently than I do when using sheet film and a hand-held meter. I simply use the exposure the meter suggests (after having tested to get an E.I. that gives me the shadow detail I want for normal scenes when I do that). With many cameras, this exposure is automatically set. This means that I'll get the shadow detail I need in every scene that doesn't fool the meter. Where the meter gets fooled is when the scene is very contrasty, especially if the meter is averaging all the values. In this case, the meter can choose a "middle value" to expose for that will severely underexpose the shadows. This is when to use exposure compensation and add an extra stop or two of exposure. You simply learn to recognize contrasty situations and add exposure. Note that for flat, low-contrast scenes you don't need to do anything. Yes, the shadows might be a bit overexposed if your meter is choosing a middle value, but that's no problem. Contrast is then expanded in the darkroom with paper contrast, etc.
Best,
Doremus