Unable to change Apple ID on my iPhone
Your iTunes ID is the same as your Apple ID. I've recently switched email addresses as well, so I can confirm that you can easily change the email address.
Assuming that you've already successfully changed your Apple ID, follow the steps that Hand-E-Food mentions:

Go to Settings

Select Store (near the bottom of the section starting with General)

Click on your (old) Apple ID

Click on Sign Out

Click on Sign In

Click on Use Existing Apple ID

Sign in with your new Apple ID email address and password
That's all there is to it!