iOS has several available gestures available on the home screen. On the iPhone, there are 5 one-finger swipes. The iPad shares these and also has a few four-finger swipes for switching applications.
Swipe Left or Right
iOS allows for more than one Home screen; iPhones have two by default. Swipe from right to left to switch to the next one. As you might imagine, swiping from left to right moves across Home screens in the other direction.
Swipe from Top to Bottom
Swiping down from the top edge of the screen displays Today View, which shows a summary, the next destination, a calendar view, reminders, and a tomorrow summary. The Today View has three sections: Today, All, and Missed. All and Missed show notifications.
Go to Settings > Notification Center to customize Today View or turn it off altogether.
Swipe from Bottom to Top
Swiping up from the bottom edge of the Home screen displays Control Center, which shows common control toggles for various settings. Toggle Airplane Mode, Wi-Fi, Bluetooth, Do Not Disturb, or Rotation Lock. In addition, adjust brightness, enable AirPlay, toggle a flashlight, and more.
Go to Settings > Control Center to set how it can be accessed.
Swipe Down from Middle
Swiping down from the middle of the Home Screen reveals Spotlight Search. Use Spotlight to find Applications, Contacts, Music, Podcasts, Videos, Notes, Events, Mail, and a whole lot more.
Go to Settings > General > Spotlight Search to set what is included in search results, and to set the order in which results are displayed. For example, you can specify that application results be returned before mail items.
iPad Multitasking Gestures
If Multitasking Gestures are enabled on iPad, a few four-finger gestures are available for switching apps. There's even a five-finger gesture to return to the Home screen.
Go to Settings > General > Multitasking Gestures to enable these gestures.
Swipe Up with Four Fingers (iPad)
On the iPad, swipe up with four fingers for Multitasking. This is an alternative to double-clicking the Home button. Multitasking allows us to switch between apps by tapping on thumbnails of recently opened programs. Alternatively, you can quit apps by flicking them up and away.
Swipe Left or Right with Four Fingers (iPad)
With four fingers on the iPad screen, swipe from right to left to switch to the last opened app. Apps switch in the order in which they appear when swiping up for multitasking. Swiping from left to right move the other direction through recently opened apps.
Swipe to Home Screen with Five Fingers (iPad)
In any app, touch five fingers to the iPad screen, then bring them together, in a grabbing motion, moving them from "open" to "closed." This gesture exits the app and returns to the Home screen; it's the same as pressing the Home button.