I just joined a new family: how do I get access to iCloud to back-up content of my iPhone?

If they are Sharing iCloud storage space and you did not have a purchased Storage tier, then you automatically start using their shared storage. You don't need to do anything.

If you are using your own Purchased Storage, then you need to go to Settings->Account(top of Settings)->Family Sharing->iCloud Storage and tap on "Use Family Storage". It will automatically cancel yours, and join theirs.

Link-> Share iCloud+ with your family - Apple Support