You can absolutely just sit down, read the classics, and then read discussions of them. You can gather ideas, and it's not impossible that you could get your own thoughts published.
However, you'll miss a lot of the point. The classics aren't a set of facts you can memorize and apply, the way you can with programming or the hard sciences. They're a set of world-views, different from your own and from other people. Even when you bring your own world view to it, even by reading what other people have thought of it, you'll still miss that dialectic. You'll miss the skill of listening and challenging your own ideas.
It would be a bit like trying to learn to dance from reading books. You can, but you really can't. You need the eye of somebody experienced to tell you what you can't see or feel for yourself because you have only your own perspective.
You can easily find yourself a book group or other discussion forum. Ideally, it would be more than just casual readers, but people who were dedicated to the domain you're studying. That might be bulletin boards or listservs or other places where scholars gather. If you're not challenging yourself against the specialists, you're going to miss out on a lot of the real insights.
Ultimately, there's no real substitute for the liberal arts education process, where you work with other students, guided by an instructor. Personally, I like the idea of Signum University, which is working on taking that idea as seriously as a brick-and-mortar college but entirely online. It's much, much cheaper, and presents the opportunity to do real scholarship.