Punch-Drunk Love - I wasn't as taken with this as alot of other people are but if you want to see Adam Sandler is a "real" movie this is your only option.
Days of Heaven - It's been a long time but I know I liked it. If you're building a classic movie collection this probably belongs.
Night of the Living Dead - This one is still pretty scary and from a commercial standpoint it was unique at the time. And of course it was a huge influence. I certainly hope this makes your cut.
Raising Arizona - I thought this was minor Coen Bros.
Once Upon a Time in America - Interestingly this is an American epic made by an Italian, about Jewish gangsters (who are usually peripheral at best in American movies). It's pretty impressive but it's not something I'd throw on causally on a weekday night, so you probably only need to buy it, as opposed to at least renting it, if you're really a fan.
Modern Times - One of the essential Chaplin movies. His last movie to be even a quasi-silent. He wouldn't have been allowed to make this after WWII.
Hot Fuzz - Not my cup of tea but it has alot of fans.
The Treasure of the Sierra Madre - This is one of the great Classic Hollywood Adventure movies and a key Bogart film. Some of these other movies in this list are more important/influential but this is one I get the most enjoyment out of.
Groundhog Day - This is pretty universally beloved, so you'll get alot of weird/angry looks if you admit to not liking it.