IOS App Store gives me “Cannot connect to the iTunes store” error message

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.

Ah, OK, I think I found a resolution. It's still not quite clear why this works or why there is a problem, but:

1. First, head to the Settings in your iPhone.

2. Click on General.

3. Next, head to Date and Time which is down the list.

4. Once you’re in there, you’ll want to change Set Automatically to Off.

5. Go to Set Date & Time.

6. Click on Today’s Date. Once you do that, you’ll be given the option to change the date.

7. Move the year one year forward to 2013.

8. This will prevent the error from happening. If you wish to change the date back to the real date, simply set Set Automatically to On and everything will return to normal.

9. Head into Passbook and marvel at the apps.

(source)

That fixes the issue on my phone.