Star Trek II: The Wrath of Khan

Essential, and the best film in the franchise by a margin nobody seriously disputes. It is also the most watchable entry point for anybody who has seen no Star Trek at all: the relationships are established in the first twenty minutes, the plot is a chase between two ships, and nothing in it requires the series.

One optional piece of homework. The villain is a character from a single episode of the original series, Space Seed, who was left on a planet by Kirk fifteen years earlier and has spent the time since watching his people die. Watching that episode first makes the film richer and is genuinely not necessary; the film restates everything you need in Khan's first scene.

What makes it work is that it is about age. Kirk is fifty, has taken a desk job, is given reading glasses as a birthday present and spends the film being told by everyone that he is avoiding something. The antagonist is a man of the same vintage consumed by a grievance he has nursed too long, and the film is explicitly asking which of the two the audience thinks has aged better. That is an unusual spine for a summer film.

The submarine-movie staging of the nebula sequence, the Ceti eel, the no-win scenario and the ending are all as good as their reputation. The Director's Cut adds around three minutes and clarifies one relationship; either version is fine.