Is the problem continuous or does it come and go? Have you tried resetting your iPad? Press and hold both the power and home buttons until the Apple logo appears, then release both buttons and wait for the iPad to restart. If that doesn't work, try resetting all settings. Go to Settings>General>Reset>Reset All Settings. This won't erase any data but you will have to re-enter any passwords for sites that normally remember you and your wallpaper will revert to the default one. If this doesn't work, your iPad may have developed a hardware fault in the touchscreen which would explain the spurious inputs that cause the strange activity you've witnessed.
If you're near enough to an Apple Store make a Genius Bar appointment to have a diagnostic done to determine if there is indeed a hardware fault. If your warranty and AppleCare+, if any, have expired there would be a repair cost involved.