A great starting point are the lecture notes for physics 219 at Caltech:
If you pick up any QM textbook that doesn't contain "no cloning theorem", "entanglement", "EPR (Einstein-Poldolsky-Rosen)", "threshold theorem", "quantum cryptography", "Shor's algorithm", or the word "Channel" in it then it is out-of-date. (Similarly, don't take a QM course that doesn't have most of these words on the sylibus.)
The quantum information community has really made abstract quantum theory interesting in the last 20 years or so. 20 years ago it would seem heretical that one could study quantum theory from an abstract point of view without specific interest in a particular physical system. Now there are at multiple papers on the quant-ph arXiv every day that do just that. The surprising fact: there are lots of interesting research problems available even in the finite-dimensional Hilbert space case. ("Interesting" here means that solutions of such problems appear quite frequently in Physical Review Letters, Journal of Mathematical Physics, Communications of Mathematical Physics, Nature, ect.)
Note: QM can get technical much faster than classical mechanics does. If you want to actually analyze atoms besides hydrogen or solids then it's going to be a quite a while before you can do it rigorously. It's analysis. Reed and Simon's "Methods of Mathematical Physics" books are a must if you want to go in that direction.
BTW, if you want to understand what a Hamiltonian is, go get a hold of Arnold's GTM book on classical mechanics. It is clearly the best.