I don't know about the other recommendations, but at least for mine, it was assigned by a prof. and I liked it. The prof. probably assigned it because they thought it was "the best" for intro quantum. So, by asking if there's a prof. that will tell you which is "the best," you aren't going to get any better answers since often students only know about texts assigned or recommended by a prof., so the students' recommendation of what is "the best" is really a trickle down of what a prof. thought was "the best."
As for the just learning the mathematical tools for QM, on an elementary level you'll need differential equations and linear algebra. At higher levels you'll need partial differential equations, Fourier analysis, perturbation theory, path integration, etc. Most of this is taught in upper level undergrad physics classes. How much you need depends on how far into QM you want to go. In my mind, a "math tools for QM" book without any QM is not worth reading; you won't learn the context of what you're doing in relation to the math or the physics. You're better off just learning the math from a QM book or a regular math text.
If you just want to go straight into string theory, there are popular science books which will introduce you to the subject. If you want the full technical details of string theory, you'll probably need
much more knowledge of physics than intro qm and relativity. I don't know any more about string theory than what's in popular shows on the Science Channel, so I can't say for sure what all the math/physics it requires.
Probably of interest to you:
http://ocw.mit.edu/courses/physics/8-251-string-theory-for-undergraduates-spring-2007/