@Inaventu- I teach children and adults how to play the piano, but I know what you're talking about when it comes to three chord songs. Students often ask me what they can possibly do with only three chords, and then I show them how many of their favorite songs only use three chords. I'll sing and play some easy songs, like "Michael Row The Boat Ashore", and they'll realize they've grown up hearing hundreds of three chord songs and didn't know it.
The next chord I'll teach after my students learn the basic I-IV-V progression is a V 7th, since that helps reinforce the pattern. A V 7th introduces a lead tone that really wants to resolve to the tonic or I chord. Once students hear that tension and resolution between V 7th and I, they start to understand how all chord progressions work, not just the traditional I-IV-V.