A good introductory text that covers QM thoroughly is Introduction to Quantum Mechanics by David J. Griffiths.
It assumes a competent math background. You'll want to have learned the equivalent of a one semester linear algebra course, a year of single variable calculus, a semester of multivariable calculus, a semester of Partial Differential Equations with a focus on Fourier Analysis and possibly some complex analysis. The class I took on PDE and Fourier Analysis was geared for prepping me for QM and the book we used was called "Partial Differential Equations and Boundary Value Problems with Applications" by Mark A. Pinsky. and we covered chapters 0-3, and 5. The book was terrible, but it's a good place to start if you want to see what kind of math you need. As for the complex analysis, you only need it for the latter parts of Griffiths book, but if you cover the equivalent of chapters 1-3.2 of Complex Variables by Stephen D. Fisher you'll be fine, you'll need to have some single and multivariable calc to understand complex analysis but nothing more.
Griffiths will try to give a cursory explanation of the PDE, complex analysis, group theory, or whatever kind of math you're dealing with in particular section of the book, but unless you have an insanely high IQ you're probably not going to be able to learn the math from Griffiths alone, which is where the above books come into play.
If you just want to see what quantum mechanics is like, and don't care for actually learning how to do it yourself, you can probably use Griffith's book with just a course of linear algebra and single variable calculus.