These films pick up several years after the conclusion of The Original Series and The Animated Series and feature the same main cast, further expanding their stories.

In multiple ways, these movies are the culmination of what The Original Series was building. Increased budgets, longer production schedules, and greater scope allowed the movies to go bigger and deeper than the show ever could. The world feels more consistent and realized. The characters (especially Kirk, Spock, and McCoy) feel more human and fleshed out. These films turned Star Trek from a series of isolated adventures into a universe.

Episodic or serial?

Mixed. Each film can be enjoyed on its own, but there is more continuity between them than there ever was between the show’s episodes. The Wrath of Khan, The Search for Spock, and The Voyage Home form a clear trilogy and The Undiscovered Country refers back to their events.

The Motion Picture and The Final Frontier are more standalone.

Idealistic or cynical?

Mostly idealistic. The greater realism and continuity mean that actions are more likely to have lasting consequences, but this is still a world where holding to utopian ideals is the right choice.

Worth watching?

While the films are overall uneven, several are excellent and absolutely worth seeking out. Also, because a film is a bigger cultural event than an individual episode of a show, some of the most significant events of the Star Trek universe take place in these films and they have a tendency to affect and be referenced by later shows.

Fortunately, the weaker films (The Motion Picture and The Final Frontier) don’t really get referenced and can be safely skipped.

Anything else that should be watched first?

You don’t need to have seen all of The Original Series, but you should see enough to have a sense of the relationships between the main characters. The one specific episode that’s the most important is “Space Seed”, which gives useful background for The Wrath of Khan.

What should be watched after this?

The natural next step is The Next Generation.

There are also three more films featuring an alternate take on these characters, but they were made considerably later.