Android - How can I prevent my phone from using any kind of internet?

Just kill the APN Connection settings. Somewhere under settings. And there is actually a switch for mobile internet. Disable that as well. Worked like a charm for me. If you can't find it, add the Power Savings widget to your home screen, it has that option

On my Galaxy S, you can hold the power button for a second and a menu will come up. Just hit "Data network mode" to deactivate it. You can reactivate it the same way.

If you don't have that, you should be able to go to Settings -> Wireless and network -> Mobile networks and uncheck Data enabled or Use packet data or similar.

I use APNDroid (Also available in the marketplace for free). You can easily turn data on or off with the press of a button.

Add the APNDroid widget to your home screen to easily turn data on or off.