Here are three suggestions for easy-to-use and extremely useful gestures that I think Apple should add to iOS with absolutely no effect on existing usage patterns:
1. Swipe up in any home screen. Currently this does nothing. What a wasted opportunity.
My suggestion for this gesture: Invoke a pre-configured action. This would allow quick access to an important app or web page of your choice, and in particular it would allow access to these without looking at the screen, a major benefit if you are driving, in a meeting. etc. This gesture should work whether you are in an 'ordinary' home screen, a 'folder' screen, the 'search' screen, the 'recent apps' screen, or the 'voice control' screen. That way, any use of the home key followed by an up-swipe would invoke the chosen action. Actions invoked would be specified in the preferences panel; perhaps the default could be 'phone' or 'camera'. This would be an alternative to my earlier suggestion for dedicated hardware buttons.
2. Swipe down in any home screen. Same as above. Note that currently in the 'search' screen, this gesture hides the keyboard, which is rarely useful in that context.
3. Press the 'home' key and tap anywhere. Currently this behaves in a way that is not useful: As you press down on the home key, nothing happens; as you then tap anywhere, the normal tap action happen but inevitably this will momentarily have to be forgotten as it is essential for you to release the home key: as you release the home key, the home page is displayed. Or, if you hang onto the home key, Voice Control is dsplayed.
My suggestion for this gesture: Universal 'get information'. This would work as followed. Almost instantly after pressing, but not releasing the home key, all tappable areas of the screen would be graphically highlighted. Tapping one of these while continuing to hold the home key would yield whatever the app desires in terms of general information. This could be help about how to use gestures on the area, statistics, etc. iOS apps should not need much help, but many apps have a non-discoverability problem, This gesture would help overcome that. If you used this gesture in a home screen, for example, this could tell you whether the app was running in the background, give you access to settings, tell you how many times you have run the app and when it was last invoked, point you to the app's web page or iTunes page, and allow you to recommend the app to others or write a review.