Hm I’m not sure what classic movie means, so I’m just going to go with the general time period (silent-1950s/60s) rather than the style (also I understand - I haven’t watched that many older or non-Hollywood movies):
- I love the original Scarface (1932). It’s pretty ugly and violent, but that in itself is interesting because it was released during the early years of the Hays Code, before it was stringently enforced, so it’s fascinating to see what kind of stuff managed to make it through before the code really took effect. Like the Hays Code prohibited depictions of “sex perversion” but there is straight-up incest in Scarface. But anyway, it also feels pretty modern and not dated at all.
- Citizen Kane (1941), I guess, I think I’m hesitant to put it down here because of the whole ~greatest movie of all time~ thing that of course is always going to color your watching it. But in addition to being important and great in the overall scheme of the development of film narrative and cinematic techniques and all that it’s actually also just a great and enjoyable movie.
- I watched Charlie Chaplin movies a lot as a kid, but my favorites of those are The Gold Rush (1925), Modern Times (1936), and The Great Dictator (1940).
- The Birds (1963) is a personal favorite of mine, of course not as iconic as other Hitchcock films, but I enjoy rewatching it, also probably partially because of a childhood nostalgia factor.
- The Manchurian Candidate (1962) and Dr. Strangelove (1964) which are two of my favorite movies of all time. They’re very different, but I watched them together, and they’re both kind of trippy (in different ways) Cold War films so I associate them with each other. Also, they’re not dated at all either, especially The Manchurian Candidate, which is really great, like they’re both very Cold War-ish, you couldn’t remove them from the historical context, but at the same time they feel modern.
- Broken Blossoms (1919) is this old D.W. Griffith film that is extremely dated, especially in its depictions of its nonwhite characters, but I think it’s a fascinating film, maybe both as a movie and also as a product of the innovations/limitations of film in that stage. I read that a lot of German Expressionists were influenced by this film like Soviet montage filmmakers were influenced by Intolerance, so that’s also something interesting to note.
- Two films that I’ve been like “OK I’m going to sit down and watch these” about for the past months: Black Narcissus (1947) and The Seventh Seal (1957).
If you’re just looking for a place to start though, I think you could just pick and choose off AFI’s great movies lists or something, just what looks interesting to you. But also remember that non-Hollywood/American films and filmmakers exist. Or look at Roger Ebert’s Great Movies list and change the filter to 1914 to ~1965 haha.. this sounds like terrible advice but you can’t really do anything to watch movies but just go and find and watch them.