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On your iPhone and iPad go to Settings > iTunes & App Store. You will find the old Apple ID there. Tap it and tap Sign Out. Then sign in to your new ID.
The Apps you have downloaded using your previous ID can't be changed to new ID. When you try to update those apps, it will still ask for old ID even if you have signed in to new ID on Settings. So you have to delete those application from your iPhone and iPad then redownload them from App Store. This is the only option and this will how Apple designed it.
To delete, tap and hold down your finger on the app and it will start wiggling. Tap the x icon over it to delete. Then go to App Store and download the apps using new ID.
If you have any questions or would like any additional information, please don't hesitate to ask.