Beware that people mean totally different levels of stuff when they say “quantum theory” or “quantum mechanics”. read more
It is possible to learn Quantum Mechanics before taking Classical Mechanics. That is what I did. But you do have to know what a vector space, basis, change of basis, linear transformation, and matrix are. Then read Dirac. You can learn the Hamiltonian point of view from Dirac. read more
First, learn basic physics (Newtonian mechanics, electromagnetism, etc.), because they are necessary for learning quantum mechanics, then learn some mathematics too (calculus I-III [single and multivariable with differentiation and integration], differential equations [ODEs], linear algebra and Fourier analysis), because that is "the language of physics" (as Leonard Susskind would say). read more