If you've previously connected to your home network and can access the internet from it, then the iPad should choose it over the cellular connection. It does not matter who your cellular provider is. It isn't up to them, and they should not be able to change this default behavior.
If your iPad is not doing this, then something is wrong and it needs to be corrected.
I could be your home network. If the iPad connects to wi-fi, but can not access the internet for an extended period of time it may switch back to the cellular network. This is especially true if you try browsing or use another app that needs internet access.
If your home network is reliable, then I'd try to Reset All Settings.
There was a problem with the early iOS 6.0 updates that affected cellular usage for some people. It was mostly in the U.K., but you should make sure you are updated to iOS 6.1.3 just to be on the safe side.
Also, bo to be on the safe side, you can go to Settings > Cellular and turn off all the Use Cellular Data for items.
If you are jailbroken (you would know) all bets are off. You'll have to take that up with the Hacking area.
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The usual fixes:
When you have problems with an app it's a good idea to clear it from RAM.
Close/Clear the app: With the problem app closed, double tap on the Home button to get the multitask bar. Tap and hold on any icon in the bar until they jiggle, then tap the minus sign on the problem app. This will remove it from RAM and insure a clean start the next time you launch the app.
If that does not work try a restart,
Restart: Hold down the power button until the power off slider appears. Slide it off. When the iPad finishes shutting down, hold down the power button again until the Apple logo appears. Apple recommends you restart the iPad at least once a month.
and then a reset if that does not work.
Reset: Hold down the power and home buttons until the Apple logo appears. The iPad will reboot. This usually takes a bit longer than a restart.
Reset All Settings: Another, somewhat more drastic thing you can do is Reset All Settings. You'll find this under Settings > General > Reset > Reset All Settings. It won't erase any of your content, apps, or accounts; but you will have to go through most of your other settings and re-tweak them back to your liking.