You have to log out of each store (iTunes, App Store, iBooks) by tapping on the Apple ID at the bottom of most tabs (Featured, Categories, etc), then log back in using the changed information.
If you use your Apple ID for email as well, MobileMe or iCloud, then you need to go into those settings in Settings > iCloud (for iCloud) or Settings > Mail, Contacts, Calendars > your MobileMe acount. It's probably easier to just delete the account and add it again with the new info.
Before you do that try logging in on the website me.com for MobileMe, or icloud.com for iCloud and make sure the new login works.