For me, one of the things I spend the extra for an Apple product is that "it just works" - iOS7 on iPad3 fails that test, and judging by the sheer volume of comments / views this thread has had, this experience is fairly widespread.
That said, the tweaks that seem to have gotten me back to a place where I can largely enjoy my iPad3 again are:
General > Reset > Reset All Settings [to get rid any of any sub-optimal settings inherited from restore etc - NB you will need to set various things up again e.g. passcode, wallpaper, wifi]
General > Accessibility > Reduce Motion > On
General > Accessibility > Increase Contracts > On
General > Background App Refresh > Background App Refresh Off
Privacy > Location Services > Turn off all the Apps you don't need
Privacy > Location Services > System Services > Turn off "Diagnostics & Usage", "Location-Based iAds", "Frequent Locations"
Privacy > Advertising > Limit Ad Tracking > Turn On
iTunes & App Store > Automatic Downloads > Turn off "Updates".
Hope it helps you.