What do I do when I get "Cannot Connect to iTunes store" when downloading an app?
What seems to work is signing out from my Apple ID (using the Apple ID link at the bottom of the App Store), then shutting off the device (hold the power button). Then I turn back on the device, go to the App store, enter my Apple ID info, and then try to download/update an app again. 

Just restarting or relogging in to my Apple ID alone didn't seem to be enough.