Bicycle Thieves - I've seen this twice. The first time was when I first started making a conscious effort to see classic movies. Then I thought it was interesting but I respected it more than I enjoyed it, when I rewatched it a few years ago it really grabbed me. This is the most essential movie on the list.
The Graduate - Good but its reputation as being one of the greatest movies of all time depends more on the way it connected with high school/college age people at the time. You should see it but it's not something I'd expect every post-baby boomer to agree belongs in a classic movie collection.
Carrie - I didn't see this until I was an adult and I still liked it. It's one of the most enduring popular horror movies or Stephen King movies in general. Or high school movies, if you look at it that way.
Stand by Me - I don't know that this is really an "important" movie aside from starting some people's careers, but it's another of the better/more popular Stephen King movies.
Inception - never seen it
The French Connection - It's good but it's enduring "greatest ever" reputation is mostly due to the car chase and being one of a slew of movies to caputer grimy '70s New York (which I guess they ruined with weird color tweaking on the blu ray?)
Duck Soup - The best Marx Brothers movies, so get this if you care about them at all.
Shane - Probably argued about by western fans at least as much as High Noon, and the arguments are more interesting. I used to be all in on this one but was disappointed last time I rewatched it. I've come around to thinking it is too self-conscious to live up to its pretensions and some of the direction seems awkward, but it's still full of great visual moments and strong performances. Apparently the sets and costumes are the most realistic of any classic western.
The Usual Suspects - I saw this once and though it was okay but overrated. I barely remember it.