I'm trying to downgrade from Catalina to Mojave, and I need to backup all of my files without backing up operating system files. Time Machine takes a backup of everything on the disk meaning that macOS Catalina files are also included. I need a way in which I can install macOS Mojave on my Mac and keep my personal data. What's the best way to backup a hard drive without Time Machine?
You can try copying all of the home folders located in /Users. After copying all the folders there to the external hard drive, you should check if all the data you want is there. After that, you can boot from macOS Recovery and erase the startup disk. After doing that, use a bootable macOS Mojave installer to reinstall macOS on the Startup Disk.
To do that, first read this on creating bootable macOS Installers: https://support.apple.com/en-us/HT201372.
To boot from that drive, you need to shutdown or restart your computer with the macOS Installer plugged in. Then, hold "Option" while the Mac is starting to boot from the installer. Once you are in, click reinstall macOS and choose your startup disk.
After macOS Mojave is installed, boot from the internal startup disk. Finish setup and create a new user. Once you are in, copy all your home directories from the SSD to the /Users directory. Then go to System Preferences and create new users and set the home directories to the home folders that were copied and remove the user that was created at the setup of macOS Mojave. Reboot the machine and you technically have your data back.
NOTE: Some compatibility issues may occur with personal files when switching from macOS Catalina to macOS Mojave.