I hear you. The first time I saw Casablanca, I thought, "It's just a bunch of cliches!" Of course, later learning they weren't cliches before Casablanca.
I especially feel sad for anyone watching the Matrix for the first time. The special effects are still copied today, so you'll never feel the awe we felt in the first scene in the theater.
That said, so many movies today rely on effects rather than story.
I recommend watching the original King Kong. Sure, it's stop motion, but it's so compelling to watch.
And, of course, The Shining, which still scares the bejeezus out of me after the 100th viewing.