Photos, contacts, and messages are all things that can be saved in iCloud. Go to Settings, then click your name at the top, then click iCloud, then make sure that Photos, Messages, and Contacts are all turned on.
Then set up your new phone as a new device and it will sync your photos, messages, and contacts (as well as anything else that you had turned on there) from iCloud automatically.
No need to mess with a backup at all.