If you only renamed your iCloud ID rather than creating a new iCloud ID, to change the ID on your phone go to Settings>iCloud, tap Delete Account, then sign back in with the new ID. This deletes the account and your iCloud data from your device, but not from iCloud. Provided you are signing back into the same account, your iCloud data will reappear on your device when you sign back in. To avoid losing photo stream photos, save them to your camera roll (if not already there) before deleting the account. To do this, open your my photo stream album, tap Edit, tap the photos, tap Share, then tap Save to Camera Roll.
If, however, you are changing to a new iCloud account with a new ID, start by saving your photo stream photos to your camera roll (unless they are already there) by opening your my photo stream album, tapping Edit, then tap all the photos you want to save, tap Share and tap Save to Camera Roll. If you are syncing Notes with iCloud you will also need to open each note and email it to yourself. You will then later have to copy and paste the text from these emails to create new notes in your new iCloud account. Once this is done, go to Settings>iCloud, scroll to the bottom, tap Delete Account and select Keep On My iPhonewhen prompted. Next, sign back in with your other iCloud ID, turn on your iCloud sync settings and when prompted about merging with iCloud, choose Merge. This will upload the data to the new account.
Changing your iCloud account will not effect your apps or photos in the camera roll or photo library.