I have Leo's mod installed so I'm not sure if this is something from the base game that I didn't know about or from the mod, but what do you have to do for the train level up requirement? It's not increasing grade, adding a specialization, or teaching. What do I do to meet the requirement if it says I have to train with them?

The train requirement is added by Leo's mod and requires that your slave have 30 learning points, which are gained from the teach action in the Meet interaction. The stat information should have been visible in the slave tab when you select the slave or in the dialogue box when you talk to the slave again.
Edit: you can also use the library to gain learning points, but that is generally very slow.

Got it, finally saw the requirement notification the slave description. I'm a dumbass. Thank you!