I do listen to a bit of rock, but it's more of bands that originated in the 90s/00s (Weezer, The Strokes, Interpol, The White Stripes, Modest Mouse, Arctic Monkeys, etc). They all fall under the umbrella of Indie music, though I think alt rock is a much better term.
Well since you started with the classics already, I'd continue with some more classic bands that you've missed. Bands such as The Doors, Sed Pistols, The Stooges, Joy Division, Creedence Clearwater Revival. You also have another option(which I'd personally advocate for) to take a dive into newer rock bands like the ones I listed in the previous paragraph. Do let me know if you're confused about what albums to start with!