When iPhone X was released, it introduced a few swipe gestures to replace the Home Button. In iOS 12, Apple brought similar gestures to iPad, even the models that still have a home button. It seems like this new way of doing things is here to stay, so I wanted to try to demystify it. The below applies to all iOS and iPadOS devices I know of, running version 12 or newer.
The basic idea is this: you take one finger, place it on the very top or bottom of the screen, and slide it toward the other end. If you're on the top, you slide down; if you're on the bottom, you slide up. You have to start from the area very close to the top or bottom edge; too far from that and iOS will assume you're just touching the screen like normal, not trying to perform one of these new commands. You'll know if you've started in the right place by the special sound iOS will make. This same sound, at higher pitches, is used to indicate what will happen during your swipe.
The first instance of the sound--heard when your finger lands on the right place--is the lowest-pitched of the three iOS will produce. If you start this gesture accidentally, or decide partway through that you don't want to complete it, just slide your finger some way left or right. You can also leave your finger in one place for a moment. This cancels the action and returns you to normal, with VoiceOver speaking what's under your finger.
If you're using an iPhone that supports haptic feedback (all modern ones do), you'll also get haptics. When you put your finger in the right place and hear the first sound, you'll get a gentle tap. As you slide your finger to the first position, you'll feel a stronger tap. When you land in the second position, this strong tap is repeated.
The Specific Commands
Let's start with the bottom of the screen. It may be easiest to find the bottom edge of the whole unit and slide up from there, as knowing where the touch-sensitive portion of the glass begins can be a bit tricky. Once you hear the first sound, keep moving your finger up. A little way along, you should hear a second sound, pitched a little higher than the initial one. If you lift your finger now, your device will go to the Home Screen. If you instead keep moving your finger toward the top of the screen, you will hear the third sound, pitched highest of the three. Lifting your finger now will bring up the App Switcher. Remember to move relatively straight up, as going too far to one side will cancel. It is also important to move your finger somewhat quickly.
Moving down from the top of the screen is similar. Find the starting point by touching the top of the screen, on or above the status bar, until you hear the lowest of the three sounds. Slide your finger down. At the second sound, you can lift your finger to open the Control Center. At the third and final sound, lifting up will open the Notification Center.
How do you remember which is which? After all, Control Center has been a swipe up, not down, since its introduction. One suggestion is to think that, since the Home button goes home and activates the App Switcher, swiping up from where the button is (or where it used to be) will perform those same tasks. As to their order, "home" has less letters than "app switcher", so is shorter. Thus, a shorter swipe is needed to activate it.
The centers are a bit easier. Once you get it in your head that both of them are now swipes down, not up, just think alphabetically (apologies to our readers who don't use English as their first language.) "Control" comes before "notification" in the English alphabet, so the Control Center is the first one you'll activate. The word length trick will also work here, as "notification" is the longer word and is also the one that needs a longer movement to open.
There's one more trick with these gestures. You can slide a finger up from the bottom of the screen, then, after you hear the first tone, slide the finger back the way it came. If you think of a similar motion to a two-finger scrub, but done with one finger and moving vertically, that may help. A comment on this guide suggested thinking of a fast swipe up, then back down, like swiping vertically to use the rotor just after moving your finger up a bit. Whatever trick you use, the gesture itself is very easy--much easier than I'm making it sound.
On iPad, this gesture will open the Dock, from where you can open a new app, open an app in Slide Over or Split View, and more. On iPhone, this gesture activates Readability, bringing the upper half of the screen down to make it easier to reach with one hand.
If you have a device with a home button, you can still use that as you always have. The usual three-finger gestures from the status bar will also still work, regardless of whether you have a home button.
Commands from external keyboards and braille displays also work as they always have. As of iOS 13, you can also customize these commands if you want to.
The last option is to customize the gestures VoiceOver uses. As with custom keyboard/braille commands, you'll need iOS 13 or newer. You can assign gestures to open the Control or Notification Center, go home, open the App Switcher, and more. Just pick come commands that aren't in use, or that you don't mind losing, and change them to do whatever you want. For instance, my own setup is a two-finger swipe up to go home, and a four-finger swipe up or down for the Control or Notification Center.