I don't remember anything from school, so please what do you say about the title?
You need at least a rudimentary understanding of classical physics in order to even understand why quantum physics is useful/important. That being said, you can learn them both at the same time.
If you are interested in physics, pick up whatever book interests you the most and start learning. When you run into concepts from other areas of physics that are necessary, you will have to take a break in order to learn them. Don't get discouraged along the way; learning physics takes a lot of effort and dedication, but the payoff is well worth it!