There wasn't any error messages.
Normally, when we buy an App, they will ask us to key in our Apple ID's password, right? So I did. And then it leads me back to my homescreen where the App will be. It started with the message "waiting..." below the App's icon, and after a while, the whole icon disappeared.
I tried a few more times after that, but it is still the same.
Do the apps show as installed on the Purchased tab in the App Store app ? If they show as installed, then have you got any restrictions set on Settings > General > Restrictions (e.g. Allowed Content > Apps) that might be hiding them ?
You could try closing the App Store app completely and then seeing if it downloads after you re-open it : from the home screen (i.e.not with the App Store app 'open' on-screen) double-click the home button to bring up the taskbar, then press and hold any of the apps on the taskbar for a couple of seconds or so until they start shaking, then press the '-' in the top left of the App Store app to close it, and touch any part of the screen above the taskbar so as to stop the shaking and close the taskbar.
You could also try a reset : press and hold both the sleep and home buttons for about 10 to 15 seconds (ignore the red slider), after which the Apple logo should appear - you won't lose any content, it's the iPad equivalent of a reboot.
And try logging out, and then back in, with your iTunes account : Settings > Store
I have tried closing the app complete and reboot last night, but it wasn't helpful.
Oh yes, I forgot to mention that, at one point, there was a message that popped up, asking me to reset my password - which I did. Still, nothing had worked.
I am actually at work now, so I will try the other options that you suggested when I get home later. Thanks!