You have your whole life to read the classics, and it would be sad to waste that by reading them when you were too young to really appreciate them, leaving you without that sense of discovery when you read them for the first time as an adult and *gasp* this is really great! This isn't a thing to hurry: most of the adult classics are best appreciated with a few years of experience under your belt. View lists of classics as "Look at all these lovely experiences yet to happen for me, I'll never be short of a classic when I feel like reading one" :) You have decades yet, and the books will still be there.

And if and when you have children, don't push them onto the adult classics. Instead, look up the canon of children's classics and enjoy yourself reading them together, scattered among newer stuff. There's no hurry.