To sum up and structure what others have said (and add my grain of salt):
a. there are a few core "quantum" concepts to learn. The normal route follows the historical one up to the 1960s (skipping too physical considerations for you), and this is indeed done well in Takhtajan's book (chapters 1 to 5). It does go through classical mechanical aspects, and requires some knowledge of undergrad math (linear algebra, Hilbert spaces, multivariate calculus, ODEs & PDEs, and a little differential geometry).
Of course be aware that this reference still does skip a lot of issues that physicists are aware of, a good physics textbook for beginners is Physics of Atoms and Molecules by Bransden and Joachain. To which one must add more recent aspects, not covered neither there or in Takhtajan, from the 1980s-1990s as in Preskill's notes. Beyond that there is field theory and ever more physics, but the basic stuff is in those three references.
b. you can then take any one mathematical aspect and push it very far. It could be the geometry, or functional analysis, or representation theory, or semiclassical limit, or complexity theory... Then it's not quantum mechanics per se anymore, but explains why some objects are labelled quantum, or studied in a certain way.
c. don't be blinded by the math if you talk to physics or engineering friends: it can look all very neat but yet a key thing to know is that most quantum systems just cannot be solved explicitely , which is why physicists introduce all sorts of approximations and asymptotics, and use computers. It's not about lack of rigor, it's dealing with non-solvable systems (e.g. only the H atom and H2+ ion can can be solved explicitely, already the He atom has a too little symmetry group to do that).
