I find this to be an interesting question because we like to peg the Klingons; Ferengi; Vulcans all as flat stereotypes, but when we look at our protagonists and the rest of humanity, we are also shown as having pretty unanimous interests. Picard likes classic literature which matches Beverly's interest in theatre and so on. We do also have characters who break from this mold, though - typically younger characters. Paris loves horror, for instance. Are Harry Kim or Wesley Shakespeare fans? Would their taste in film lean more to action films?
I digress, though. Back to movies!
I'm taking this from the viewpoint of a first showing, which would be most of the crew's first film experiences. For that reason I'm going easy on gore - I'm not sure how much your average Joe Starfleet would appreciate bloody violence (By which I mean gratuitous splatterhouse gore).
Okay, first off: "Vertigo". The Hitchcock classic, clocking in at just over two hours long which is a plus for a first-time viewer. This film is notable for its pioneering cinematography and stage direction - lending itself as a uniquely movie-based achievement. It is also in colour, which is a bonus. This is an intellectual experience to appeal to your Picard-type lover of literature.
"The Godfather". I was going to put this as the first movie, but its three-hour runtime I thought could be initially off-putting. This is the ultimate crime drama held up by its performances. "The Godfather" wasn't afraid to make you squeamish - The horse head; the strangling - but everything it did, it did with narrative purpose.
After the deep "Vertigo" and "The Godfather", I wanted to throw a Disney movie in there. It's a hard choice, but I think the best option is "Inside Out". It edges out "The Lion King" due to its pretty mature themes which appeal strongly to adults. The movie deals with depression, loneliness, childhood and its abandonment. "Inside Out" is also a technical marvel to behold (Although I'm sure the Enterprise computer could create something more beautiful in less than a minute).
War movies are an incredibly popular genre. I originally wanted to go with "Saving Private Ryan", but I think that would break my gore rule. Instead I'll go for "First Blood" which serves as a good "solely entertainment" movie. You root for Rambo as he tries to survive. It does contain some gore, but comparatively little to some of the alternatives. It is also loosely historical, with Rambo being a Vietnam vet suffering from PTSD.
Okay - on to comedy. At first my intention was to fill this spot with one of the "Three flavours Cornetto" movies, but all three of those kind of break the gore rule. Instead I'll go for "Monty Python and the Holy Grail". Like one of the Cornetto movies the comedy is pretty timeless which is essential for showing someone hundreds of years in the future. It has some of the most famous comedy routines in film history and it also demonstrates the ability of a low-budget movie.
Two left. I want to include a superhero movie since they currently dominate the market. After some thought, I think I'll go with "Batman Begins". Since it is unclear whether superheroes are still a thing in the 24th century, I thought it would be a good idea to go with an origin story. While You could see it as a "turn your brain off" kind of movie, "Batman Begins" actually has some good themes, such as- ...Who am I kidding, it's a bat-ninja beating up criminals, bur its awesome.
My last thought, since they seem the most popular whenever film has ever been shown in Trek: a horror movie. For this I'll go with 1958's "Dracula". Unlike the previous version, it is in colour (plus) and it has Christopher Lee as the titular baddie. It is also based on a popular book I'm sure much of its audience would have read.
So those are my ideas. Do you disagree with any of my reasoning? I'd love to hear your suggestions as well.