Strictly speaking, you need grapes to make wine (this ignores fruit wines, for simplicity).
Each item in a wine kit has a purpose:
Kits are pasteurized, so there's no natural yeast. When making wine from fresh fruit, add commercial yeast so you're not at the whim of whatever yeast is growing on the grape skins. Commercial yeasts also typically have a "killer factor" that helps them beat out wild yeasts and bacteria. Some folks go natural on yeast ... but that's a risk I am not interested in taking.
Potassium metabisulphite helps stunt wild yeasts and bacteria, and in lesser quantities doesn't harm commercial yeast. It is also an antioxidant and preservative, lengthening the shelf life of your wine. In my experience, commercial wines made without sulfites have a shorter life span.
Fining agents including kieselsol, chitosan, and bentonite help clear the wine faster. Time will clear the wine, but the fining agents produce a quicker result and may produce a better result. Commercial wineries use fining agents -- in Bordeaux it was common to use egg white to fine the reds. Now days there's a lot of filtering and some wineries are using centrifuges.
Sorbate is essential if you are sweetening the wine, unless you are using a sterile filter (removes all yeast) or adding sufficient alcohol to raise the level above what the yeast can survive. Kits include sorbate on all wines (dry wines don't need it) to avoid problems with customers who don't follow the instructions and let the wine ferment to dryness. If the wine is dry (SG below 0.996) you can skip it.
If no one mentioned it, sorbate will prevent a renewed fermentation in a wine that is not actively fermenting. It does not stop fermentation, nor does sulfite.
I recommend starting with a kit so you can focus on the mechanics of wine making without having to figure things out. After you do a few kits, it all will make better sense.
When making a kit -- follow the instructions. Do not get inventive until you know what you're doing. Follow the steps in order, keeping in mind that all time frames are a minimum, and your hydrometer tells you when things need to happen. The only thing that may be shorter is the initial fermentation -- I've had wines ferment to 0.990 in 3-4 days (although that is the exception, not the rule). Letting the wine age an extra week or two at most steps hurts nothing.
When making wine from scratch? Think about what you're doing, and do what you're comfortable with. As was recommended, keep good notes and record SG EVERY TIME you touch it.