Which movie to see depends on the personal taste and mood, but there are films that one cannot miss:
Casablanca (1942)
2001: A Space Odyssey (Stanley Kubrick, 1968)
Goldfinger (1964) and the other 007 movies with Sean Connery starring as James Bond
North by Northwest (1959) or other films by Alfred Hitchcock
Blade Runner (Ridley Scott, 1982)
The Dollars Trilogy by Sergio Leone
The Star Wars trilogy/hexalogy by George Lucas
Close Encounters of the Third Kind (Steven Spielberg, 1977)
And a few more amongst my favourites:
The Omega Man (1971) -- watch this one instead of the crappy remake I Am Legend (2007)
Planet of the Apes (1968)
The Matrix (1999)
Barry Lyndon (Stanley Kubrick, 1975)
The Lord of the Rings trilogy (2001-2003) by Peter Jackson
Everything You Always Wanted to Know About Sex - But Were Afraid to Ask (1972) and almost any other Woody Allen film, with preference for the old ones
Groundhog Day (1993)
The Pink Panther (Blake Edwards, 1963) and the other movies with inspector Clouseau / Peter Sellers
Breakfast at Tiffany's (1971)
Saving Private Ryan (Steven Spielberg, 1998)
Seven Samurai (Akira Kurosawa, 1954)