I don't know the appropriate setup to encrypt whole disk drive using FileVault 2 and Automatically backup it without issues. Currently, I'm using a 2nd-gen Intel i7 MBP running macOS High Sierra 10.13.6 (17G5019), One Internal Crucial SSD, APFS volume, and using Time Machine for Backup on a G-technology External Drive, Mac OS Extended (Journaled).
/dev/disk0 (internal, physical):
#: TYPE NAME SIZE IDENTIFIER
0: GUID_partition_scheme *500.1 GB disk0
1: EFI EFI 209.7 MB disk0s1
2: Apple_APFS Container disk1 499.9 GB disk0s2
/dev/disk1 (synthesized):
#: TYPE NAME SIZE IDENTIFIER
0: APFS Container Scheme - +499.9 GB disk1
Physical Store disk0s2
1: APFS Volume Macintosh HD 456.0 GB disk1s1
2: APFS Volume Preboot 20.4 MB disk1s2
3: APFS Volume Recovery 512.0 MB disk1s3
4: APFS Volume VM 10.7 GB disk1s4
/dev/disk2 (external, physical):
#: TYPE NAME SIZE IDENTIFIER
0: GUID_partition_scheme *4.0 TB disk2
1: EFI EFI 209.7 MB disk2s1
2: Apple_HFS G-DRIVE USB 4.0 TB disk2s2
3: Microsoft Basic Data G-UTILITIES 4.1 GB disk2s3
Best Answer
You can enable FileVault 2 from System Preferences > Security & Privacy > FileVault. Just press "Turn on FileVault..." and full disk encryption will be setup.
Full Disk Encryption in this sense means that all your own data on the Macintosh HD file system will be encrypted. There is a preboot partition that isn't encrypted, as this is where the software necessary to decrypt the drive is stored. Without it, you wouldn't be able to boot the Mac.
When you plugin the USB drive, macOS will ask you if you want to use it for backups. If you confirm that, it will ask you whether or not you want the backups encrypted. Set the checkmark and encryption of the backup will automatically be setup for you.
If you have already started backing up to the external disk without encryption, you can enable encryption afterwards like this:
Open System Preferences > Time Machine. Click "Select Disk..." and remove the external disk.
Do it again and select the disk again, and it will ask again for encryption. Set the checkmark and you'll be asked to set a password. The disk will then be encrypted. The backups already stored on the drive will be preserved.