Hi Corey,
Your instructor's developing sheet looks almost identical to the one I had in my darkroom class. When I asked my instructor about whether those times would work with push or pulled film, or other films, he said yes (which I knew even then couldn't be right.) We weren't forced to learn the zone system, which I think is impossible anyway if you are shooting roll film (unless EVERY image is under the same lighting conditions, which is rare, unless you are doing studio work). That said, I dutifully bought the bible in three parts, aka Adams trilogy of books. I tried reading them multiple times over the years, but they always lead to the same result - me falling asleep! He's not an easy read and most of what he's talking about refers to shooting sheet film, which is not what you are doing. Now, of course, if this is something you have to learn for your class, then you just gotta learn it, but perhaps you can ask your instructor by what he means when it comes to applying the zone system to roll film. Or is he talking about having a wide ranges of tones in the print? The two are not the same.
As a beginner, remember to KISS (keep it simple, stupid). Choose one film. Shoot it at box speed at the developer recommendations (which you are probably doing). Thousands, perhaps millions of people did that throughout the years and had good images. That said, you should also take a roll and bracket - normal exposure, 1&2 stops over, and 1&2 stops under. Do it in different lighting conditions - bright, contrasty light, dull, low contrast light, indoors, snow scenes, etc. You can try seven different situations on one roll of film. From that you should discover what works best for you, and if you should shoot at box speed (say, for dull light), or at half box speed for contrasty light.
Similarly, when you are in the darkroom, don't be afraid to use as much paper as you need to get it right. Like film, don't be afraid to push the limits of your printing - try to make the best print you can at different filter grades. You may find that you like more contrasty prints, or perhaps less contrasty prints. (No need to do this all the time, but someone suggested it to me, and it really helped me out.) And, of course, you may have to consider what your instructor likes. I find that men (in general) print far darker and/or more contrastier than I do, not that I am representative of all female photographers.
Of course, everyone here is giving you opinions on what to do based on their own experiences, so you have a lot to filter through. Since I am no different, I will give you two of my recommended books that helped me a lot. In addition to some of the books listed above, I really love these two books:
1- Way Beyond Monochrome, 2nd ed. By Ralph Lambrecht (he has already replied to this thread) and Chris Woodhouse. Lots of advice for film and printing.
2-The Master Photographer's Master Printing Course by Tim Rudman (he used to be on APUG a while back, but not so much anymore). Maybe you can find it used or in the library.
Cheers.