Do you have to transfer apps from an old phone to a new one?
When setting up a new phone through icloud, is there a way so all of the apps don't get installed but I keep other things such as photos, contacts, and messages?
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If your photos, contacts, and messages were synced (e.g., iCloud Photos rather than iCloud Backup), you can set the phone up manually (i.e., not using Quick Start or a backup) and then turn those same syncing options on in the new phone and those things will sync with it.
If there are any previously purchased apps that you want to download, you can go to App Store > user icon near top right of screen > Purchased and download them