You may not need to pay the monthly fee. You get 5GB free.
If you just set up a new Apple ID for iCloud and the only thing you have done with it is subscribe to more iCloud Storage, I would recommend that you downgrade back to the 5GB free level for that Apple ID so you won't be charged for it in the future. See iCloud storage upgrades and downgrades - Apple Support
Then you can upgrade the storage on your original Apple ID (if you need to, but you may not as 5GB is a lot).
Then go to Settings > iCloud and turn on backup and it should backup to your original Apple ID automatically while it is plugged in to charge if connected to wi-fi and you can also force it to backup with the backup now button. Be aware that the contents of the backup will not be visible if you use a computer and log in to iCloud.com. A backup can only be used to restore an existing iOS device or set up a new one. The things that are visible at iCloud.com such as Contacts are put there when you turn on Settings > iCloud > Contacts for example and any changes made later are automatically synced immediately (no backup required). Photos are synced to iCloud.com when you turn on Settings > iCloud > Photos > iCloud Photo Library and they may take a long time to sync.