If you have photos in photo stream that aren't in your camera roll and that you haven't backed up anywhere else you will need to save these to your camera roll first. To do this, open your photo stream album on your iPhone in the thumbnail view, tap Edit, tap all the photos to select them, tap Share, then tap Save to Camera Roll. When you move to the new iCloud account (ID) they won't be in your photo stream, but you will still have them in your camera roll.
After doing so go to Settings>iCloud on your phone and turn all data you are syncing with iCloud (contacts, calendars, etc.) to Off. When prompted choose to keep the data on your iPhone. (If you are syncing iWork documents with iCloud, also open your iWork apps and turn off iCloud syncing and choose to keep the documents on your phone.) After everything is turned off, scroll to the bottom and tap Delete Account. Next, set up a new iCloud account using your other ID, then turn syncing for your data (contacts, etc.) back to On. When prompted, choose Merge. This will upload the data to your new account.