Short of cleaning up your iPad of excess data stored on it, there is one method the Genius Bar recommended that works very well.
The Location Services setting on the Settings App if turned on, will run in the background without any notification in the active Apps taskbar. If you disable Location Services it can free up a lot of space for most applications to run without issue. So unless you need Location Services on for specific mapping needs, turning it off can solve most quitting issues.
The other most common way to solve quitting, is to swipe the application off the screen after hitting the home button to navigate the open application. The home button usually needs to be double tapped to bring up all open applications. Then horizontally swipe to navigate between applications, and swipe up to remove the frontmost application.
Any other solutions people are welcome to comment or edit this tip.