Two very different Alien films set the tone
The Alien series really has two founding films with different identities. The first is a slow, claustrophobic horror film. The sequel is a bigger, louder action film. Most of the debate about the series is really a debate about which of those two you prefer.
Alien (the original)
The original Alien is a haunted-house film set on a spaceship, with one creature and a tightening atmosphere. It is the best place to start because everything after it either extends or reacts to it. It is also the purest horror film in the series.
Aliens (the sequel)
Aliens turns the horror into action, adding more creatures, soldiers and gunfire. If you found the first film too slow, this is the counterpoint. It is widely considered the other high point of the franchise.
Alien 3 and Alien: Resurrection
Alien 3 strips the series back toward a smaller, grimmer story; Alien: Resurrection pushes it in a more stylised, almost comic direction. Both are divisive. Watch them to complete the original arc, but know they split opinion.
Alien: Romulus
Alien: Romulus is a recent return that blends the horror of the first film with callbacks across the series. It is a strong modern entry point for someone who wants a current Alien film without committing to the older ones.
The crossover: Predator
Predator shares the Alien universe in expanded media, and fans of the creature-hunting setup will find a similar tension. It is not a direct sequel, but it is the natural companion watch.
A suggested order
If you want the core experience: watch Alien, then Aliens, then Alien: Romulus. If you want the full story after that, continue with Alien 3 and Alien: Resurrection. Begin with the original — every other film makes more sense once you have seen what started it.