Yeah, you really won't understand either string theory or loop quantum gravity at all (specifically, you won't understand the key differences) if you don't know quantum mechanics.
If you want an over-simplistic idea of the difference between them, string theory tries to start from a non-general-relativistic spacetime, build a theory there, and then extend to general-relativistic spacetime, while loop quantum gravity redefines space and time altogether in order to get general relativity to quantize the way quantum mechanics demands. One starts from quantum mechanics and heads towards general relativity, one starts from general relativity and heads towards quantum mechanics.
Since I don't know much of the direct physics involved, I'll just get you started with some differential geometry, which is my specialty:
Two sets of notes on differential geometry that look decent; I haven't had the time to go through them, and I unfortunately can't recommend any books because all my math classes have been bookless (lecture notes only)
Wait, I can recommend a book! My favorite book, in fact: Road to Reality, by Roger Penrose. Note: he much prefers loop quantum gravity over string theory, so towards the end of the book he's a bit biased. But it's a good framework for figuring out what stuff you'll want to know and it is fairly informative. Moreover, it's highly mathematical, unlike the elegant universe.
A general guide to the math/physics topics you would want to look at would be something along the lines of:
Newtonian dynamics, multivariable calculus/linear algebra, something on waves
Classical Hamiltonian/Lagrangian mechanics, classical electromagnetism, intro differential geometry (diffeomorphisms, smooth structures, bundles bundles bundles), intro group theory (with representation theory), intro real analysis, intro complex analysis, point-set topology
Intro quantum mechanics (non-relativistic), more Lagrangian mechanics, Lie groups/algebras, into algebraic topology, intro differential topology, intro algebraic geometry
More quantum mechanics (relativistic), more algebraic topology (homotopy theory, various homologies), more differential topology (symplectic forms, de Rham cohomology, etc), graded algebras, functional analysis, multivariable complex analysis, more Lie groups/algebras (spinors, lots of spinors), more algebraic geometry (modular forms, holomorphic curves)
Quantum field theory, the Standard Model, lots of algebraic topology (tons of homology), lots of differential geometry (Chern classes, Kahler manifolds, Seiberg-Witten invariants, Morse theory), superalgebras, lots of analysis of all kinds
To give you an idea of the time-scale of these things, I'm about at the last set of things I mention, having taken a bunch of 1st/2nd year graduate math courses.
