The Godfather (1972) → The Godfather Part II (1974)
This is the most basic kind of good sequel: another great story, filmed solidly, to follow up a great original. It's actually a bookend to the original, with the scenes divided between Vito Corleone's back story and those chronicling Michael's downfall, but it qualifies as a sequel in the most basic sense that it continues the story begun in the original film.
While the first movie satisfyingly ended with Michael symbolically becoming his father despite his protestations to Kate in the first act, from the perspective of the combined saga (let's just do the right thing at the beginning and pretend Part III had never been made) it merely foreshadows his irreversible destruction of the Corleone family and the profound emptiness resulting from his supposed final triumph over his enemies.
It speaks to the power of the sequel that it's hard for me to watch the first movie now without thinking about the ultimate fates of the characters around Michael.