Brewing

This is far from the only guide to brewing, but it's the one you're at now. Let's not second guess the decision. Here's an Amazon List which includes the following essential items:

Bucket

Don't underestimate it. It'll catch spills, let you rinse stuff, and help you keep the gear tidy when not in use.

Apple Juice, 1 gallon

I'd like to see you try to make cider without it!

Korkens

Technically called "swing top bottles", but in my heart the IKEA Korken sets the bar. If the cider keeps brewing when it's not supposed to, the lid will pop open. This is preferable to the glass popping open.

Pump

When it's time to move the cider from one container to another, you want to leve behind the sediment. This makes that possible.

Hydrometer

It's worthwhile to know how much alcohol your alcohol has.

Cleaner

Brewing is about keeping one and only one microbe alive. You don't want the yeast you paid for to be competing with others.

First Timer's Kit

Speaking of yeast, it's included in this kit along with nutrient, a carboy, and an airlock.

Funnel

The pump works to move fluid, but boy oh boy are you not gonna have the patience to use it every time.

Now, arguably, you could get by with just the juice and the kit, if you plan on drinking straight from the carboy and letting the sediment float around in it.But don't argue that, because you're better than that. Instead, do the following.

Your First Brew

1. Fill the carboy with water.
2. Place the carboy, disassembled airlock parts, pump, korkens, hydrometer, and funnel in the bucket.
3. Add one tablespoon of cleaner to the carboy. I didn't include a tablespoon measure on the shopping list, but this mesure doesn't need to be exact. Good luck, I believe in you.
4. Shake the carboy thoroughly. My preferred technique is to keep my right palm over the openning with my right index finger through the handle. Then I lift the base with my left hand and shake it all about.
5. Empty the carboy into the bucket. There may be residue in the carboy. Add water, shake, and empty until there is no residue.
6. Get each tool surface thoroughly engaged with the cleaning solution. To get the inside of the pump, you'll have to pump solution through it, from the bucket to the korkens. Fill them and set them aside.
7. After everything has had a minute or two to air out (except the korkens) use the funnel to pitch the juice into the carboy. Stop about halfway through.
8. Add the entire yeast packet, then use it to measure out an equal amount of nutrient. This is more than you need, but that's okay for your first brew. Experiment with amounts in subsequent batches.
9. Shake the carboy again. Be careful when removing your palm- You should already have some pressure building up. Not much, but enough to spew some wort around (that's the name for the solution you've prepared in the carboy). Try to vent it into your hand to make clean up easier.
10. Using the funnel again, pitch in the rest of the juice.
11. Assemble the airlock. Use cleaner to fill it to the water line. Put the airlock on the carboy. As you press the carboy into place, notice the water level shift from the pressure. Soon, it will bubble. The produced CO2 can escape, and fresh air cannot enter. Set the carboy up someplace cool and dark, and keep an eye on the rate at which the airlock bubbles.
12. Wait a couple weeks. What you are actually waiting for is for the yeast to die. It consumes sugar and converts it into more yeast, CO2, and alcohol. Eventually there is so much alcohol the new generations of yeast die. Or they could die from heat (but they shouldn't bc we kept it cool) or from competing microbes (but they shouldn't bc we cleansed everything) or from lack of sugar or other nutrients (but they shouldn't bc juice has a lot and we added nutrient). The ideal death here is that they did such a good job making alcohol they sanitized themselves out of existence.
13. Rack the brew. You should have a layer of sediment (dead yeast, mostly) along the bottom. It's harmless, but icky. Remove the filter from the pump (because even if you manage to fit it *in* the carboy you won't be able to get it *out*) and pump the brew into the korkens. Control the depth of the pump intake to avoid pulling up sediment.
14. Notice that you haven't used the hydrometer. It wasn't for this, your first brew. Just drink it. On the next batch we'll start blooming yeast, measuring specific gravity, labeling things. But not this time. This time, enjoy your well deserved brew.