To learn Korean as a K-drama fan, the real game changer is combining a structured app with authentic material. Duolingo is exactly what you said - way too shallow if you actually want to understand what's happening on screen. What works better is starting with HelloTalk or Tandem, where you chat with native Korean speakers in real time - it's not passive like a traditional app, and you learn the expressions people actually use in dramas, not just textbook stuff. Then you add Memrise on top, which has way denser courses than Duolingo with real contexts, not just random words.
Whoever suggested Anki is right that it works, but starting with that is like jumping straight to climbing a mountain. Anki is boring at first, yeah, but it becomes essential once you've already got the basics down. First you build a solid foundation with an app that has actual grammar structure - I'd say Pimsleur or Busuu, they cost something but have real teachers behind them and decent grammar follow-up. Then, once you don't feel completely lost, you start using Anki with well-made decks (like the ones from HangeulClass on Reddit), because at that point the material actually makes sense.
While you're learning, watch your dramas with Korean subtitles, not Italian ones - your ear needs to get used to it. On screen you'll see words you just studied and they stick in your head way more compared to any abstract exercise. It takes time, at least a few months before you can catch full sentences without reading, but it's the method that actually works for someone starting from zero with a language this different from Italian.