Genre
Science Fiction; Adventure; Parable
Here's a controversial opinion: Star Trek is science fiction. Gasp.
We're being sarcastic, of course—the Star Trek series is an iconic example of hard science fiction, with its emphasis on real scientific ideas rather than fantastic ones. That being said, this movie does involve a quest for God, so take all of that with a grain of salt.
The other main ingredient in the Star Trek recipe is adventure. Since the original series, Star Trek has sported heavy influence from classic adventure stories, in particular those of the swashbuckling and Western variety. At this point in the series, that influence is mostly present in the character of Kirk, who's still as adventurous and freewheeling as any pirate captain of yore.
Finally, The Final Frontier features aspects of a parable. Though used less prominently here, parable has always been a part of the Star Trek series. Many episodes and films create elaborate metaphors for current-day issues, hiding social commentary beneath sci-fi complexity. That's certainly present here: The Final Frontier uses Sybok's quest to analyze the terrestrial evangelical religious movements of the day.