The YouTube app is no longer auto installed on the iPad once you have iOS 6 running. You'll have to download an app from the App Store (Google has a YouTube app or there is a new app called Jasmine that's getting good reviews). Or, you can access YouTube from the browser (Safari).
The others are probably running slowly as there is a lot of demand on both the App Store and iTunes (as Apple's servers strive to keep up with the new iOS and people accessing them with their new iPhones). All I can suggest is a bit of patience as the new wave of iDevice owners settles down.
Marilyn