@TaliaRinger not that I'm qualified, but I'd start with giving lots of examples like adjunctions between presheaf categories, free/forgetful functors for monoids, groups, vector spaces, ... , sheafification, pullback, pushforward if they know algebraic geometry, etc.
After that I'd discuss properties of adjunctions and let the students figure out that adjunctions explain a lot of properties (preservation of limits/colimits, "free" constructions, ... ) that they've studied in isolation