How to Set Recovery Mode in SQL Server Always On Availability Group without downtime?
Availability groups require databases to be in full recovery mode (see Availability Database Prerequisites and Restrictions), so there should be no need to switch it to full prior to running a backup.  The database should not be switched out of full recovery mode if it is in an availability group.

As to where to run the backup, on the primary or secondary, it is a matter of  load distribution, and currency.  If you don't want the load of the backup to run on the primary, then you'll need to run it on the secondary.  If they are in asynchronous replication, the backup may not contain all of the latest data if some of the database replication is backed up in the log send queue.