To change your iCloud 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.