New iPhone 5 but no computer with ITunes?

How do I connect to the App Store when I don't have a computer with ITunes? My phone is brand new, never been used before. When I go to the App Store on my phone it says " can't connect to App Store"
Note: I have a surface tablet so it won't let me download iTunes

If you have an apple account, then you have a cloud account. Go to your settings, scroll down to iTunes and App Store, then make sure that you are logged into your Apple ID (your account). That ought to be enough for you to access the App store and any iTunes you have.

The surface tablet should let you download iTunes as it is considered a desktop PC in a way. But I'm afraid that is just your internet connection most likely. Maybe try hard reseting your phone by holding down the power button and home button together until the phone switches off. (PS this won't wipe your phone)