Square symmetric matrices (or rather complex Hermitian ones) represent the observables of a quantum mechanical system. Their eigenvalues represent the possible observed values in ideal experiments. There is a basis of orthonormal eigenvalues, which allows you to write any state vector as a linear combination (superposition) of basis vectors. Squared absolute values of inner products define probabilities. Then one needs functions of matrices, in particular the matrix exponential, which gives the dynamics of a system, and an explicit solution of the Schroedinger equation in the case of an n-level system.
Thus you need to learn enough to be able to have a good grasp on these concepts:
matrix, transpose, conjugate transpose, linear combination, basis, eigenvalue, eigenvector, inner product, matrix power series, matrix exponential. Wikipedia has good summarizing articles on each of these topics, to help you give an overview. You can skip other stuff, and get back to it in case you need it.
In analysis you need systems of linear differential equations with constant coefficients (these are related to the matrix exponential), and the Fourier transform. The latter involves integration in 3 dimensions, but again, you can skip a lot and turn back to things skipped once you need them.
Then you can look into various quantum mechanical texts or lecture notes,
for example my online book http://lanl.arxiv.org/abs/0810.1019 - the first chapter of it should be understandable even with little prior knowledge, if you can tentatively accept concepts without a full understanding. My FAQ http://arnold-neumaier.at/physfaq/physics-faq.html might also be of help. By reading these and noting where you lose track you can find out what other concepts you need to make sense of your reading. This will tell you what else you need to learn. Ultimately, almost all of linear algebra and analysis is useful in quantum mechanics, but what and when depends on what you are interested in.