Anders,
Before you start applying the Zone System, you should make sure your camera is working properly. Are the shutter speeds accurate? Are you using one lens or many lenses? Each lens will give you a different exposure. What film are you using? Different films react differently to filters, development and even long exposures. What developing chemicals are you using? Learn how that film reacts to that developer and learn how to manipulate the development with that particular developer/film combination. So the very first thing you need to do, is check your equipment. Many cameras have issues with the correct shutter speeds or even sticking shutters at slow speeds. I was told by one camera tech, that he has seen exposures up to 3 f/stops off on fast 1/250 or 1/500 sec shutter speeds. That surprised me, as usually one can tell when the slow speeds are off, but seldom can one tell when the faster speeds are off.
Nothing will disappoint you more than having bad results because your equipment is not working properly.
Good luck with your photography,
Jose