No. You should follow an established curriculum because the textbooks are written that way, such as where mathematical techniques are introduced.

Memorizing enables fast recall. If you have to know it, you'll have to memorize it even if you can derive it. And there is very little to memorize in physics. You have to know that metals are ductile, but that's not a lot of information; you're not going to check it by going back to quantum mechanics. In principle, you could use QM to derive a quantitative version, but it's computationally intractable.

In the direct relation between quantum and classical mechanics, QM is simply more complicated: you generally start with the classical laws and modify them, so they are a prerequisite. I think that there is a recent QM textbook by quantum computing researchers that get to quantum weirdness with very few prerequisites. This sounds like a good place to start, but if you want to cover the whole thing, you'll need classical mechanics.