You can save a lot of time by intelligent scrolling. Instead of dragging you can flick, to scroll quickly.
To get a the top of a page immediately, tap the Status Bar at the top of the screen. This is especially handy in Messages and Safari, for example when you decide to call the person who just sent you a text message.
If you type or copy and paste text, shake the phone and a menu will appear on the screen offering to Undo Typing or cancel. Tap Undo Typing to undo whatever text you just typed.
If you change your mind, you can restore the text by shaking again and tapping the Redo Typing button.
You can use a three or four finger swipe up to close up to three apps at once. The three finger swipe works well on an iPhone or even an iPad mini.
Pinch with four fingers to exit an app and return to the Home screen (like clicking the Home button)
You can take a screenshot at any time: Press and release the On⁄Off button and the Home button at the same time. The screenshot is added to your Camera Roll album.
By quadrupple tapping while typing you can select a full paragraph. You can then delete a whole section or copy and paste between apps
Mail now has pull to refresh (like Twitter, Facebook, etc)
Tap and hold the New Message button inside Mail to open the Drafts folder.
By keeping the back or forward button pressed the browser history pops up.
Most people use the swipe gesture to move left and right between pages of apps but:
you can also tap next to the dots at the bottom of the screen.
This works wherever you see the dots.