In SQL*Plus you can configure it to automatically run login scripts, typically used to configure your session. These are stored in the files glogin.sql and login.sql. Some examples that you'd find in these files are statements like: set serveroutput on. You can read more about login scripts here.
SQLcl also handles login scripts, however it may require some additional configuration. The easiest way to leverage this is to create a file called login.sql in your current directory, then call SQLcl from that directory.
If you're like me and launch SQLcl from many directories that approach won't work. Instead, you can create one login.sql file and have it automatically referenced. To do this create login.sql in a folder (in my case it was ~/Documents/Oracle) and then add the following to ~/.bash_profile: export SQLPATH=~/Documents/Oracle/
Now when you run SQLcl anywhere it will reference your login.sql file.
Thanks to Kris Rice and Jeff Smith for helping with this.