How do I get what I backed up onto my new phone?

When you're setting up your iPhone there should be an option to restore from a backup.

Here's an article --> Restore your iPhone, iPad, or iPod touch from a backup - Apple Support

See the following Apple Support article for information on using an iCloud Backup to setup a new iPhone: Transfer data from your previous iOS or iPadOS device to your new iPhone or iPad - Apple Support

Another option is to use Quick Start which is also described in the same support article.