I second what Nicholas has advised.
The ZS assumes you'll be basing exposure on a shadow value and then seeing where the other values "fall." Read/learn about Zones II through V so you can place a given shadow value in the approximate Zone you want it. Inky black, Zone I or II, black with texture, but still very dark, Zone III, luminous shadows in landscapes, Zone IV, Shadows on snow or white sand, Zone V, etc. And, you'll need to be able to easily and accurately meter shadows for exposure determination and the other values to see where they fall. If you don't have a spot meter, maybe the ZS is not for you.
Don't bother with film-speed testing; simply rate your film 2/3-stop slower than box speed. This compensates for the Zone VI metering technique and assumed speed point. It will be way close enough to get started with and most likely not have to be changed.
Find yourself a N (normal) developing time by photographing a scene with a full distribution of values from Zone III through Zone VIII (textured black through textured white). Make a few negatives of the scene, develop one at the manufacturer's recommended time in whatever developer you choose and make your best print on the paper you use at an intermediate contrast setting (your N). I, like Nicholas, recommend you use grade/filtration 2.5 or 3 (I like 3) for your "normal" grade. Make a straight print with the Zone VIII textured whites in the right place. If the shadows are too dark, develop the next negative 15% less than the first and try again. If the shadows aren't dark enough, develop 15% more and try again. Within about three prints, you'll have a good N developing time. (Remember, you just need to be close, i.e., within half a grade or so, so don't obsess with getting a perfect print from a straight print at a selected grade - you just need to be inside the goalposts, not exactly in the middle).
Now you have your N development time and a usable E.I. Go out and make photographs, keeping good notes. The real learning curve of the Zone System is learning how to visualize what you're going to get in the final print from the meter readings you take in the field, so keep track of those. Imagine how you want the shadows, mid-tones and highlights to be rendered according to the meter readings and see if they match what actually happens when you make a print. You'll have to adjust your visualization to agree with the realities of the medium; that's the real learning curve.
With experience, you'll know what your going to get before you release the shutter, where printing problems will be, what you need to dodge and burn, etc. And, if you find your shadows are not exposed enough for your taste, you can adjust E.I. or change development times if your negative contrast isn't getting you in the ballpark most of the time. Field notes help here.
As far as contractions and expansions are concerned (N- and N+ development times): Most contrast control can be done with VC papers these days, but there are still times you need to contract or expand development. However, you don't need to aim exactly for a particular paper grade/contrast filtration (as was necessary with graded papers in the past). So, just think that N-1 is about 20% less development than N and N+1 is about 20% more (again, you can adjust these later if needed).
Scenes that in the classic ZS would require N-1 or N+1 development can be dealt with easily by developing at N and using a different contrast filtration to make up the difference. When you run into a scene that the classic ZS would call N-2 or N-3, just develop N-1 and deal with the rest of the contrast control with your VC paper range (e.g., N-1 development with a #1 filter when your N is a #3 filter = N-3). The same with expansions; for N+2 and N+3 situations, just develop N+1 and use higher-contrast filtration to make up the difference.
That will get you more than started. Remember, if you don't use the visualization aspect of the ZS, you aren't really using the ZS, just ensuring adequate exposure and a negative with a usable contrast range. Being able to see in your mind's eye what you're results will be before making the exposure is the real advantage.
Best,
Doremus