Well, in order to understand topological strings in depth the question is more like what are the mathematical prerequisites you need.
The starting point, for a physicist, is a 2d non-linear sigma model, thus you should have some small familiarity with 2d quantum field theories. If you have studied all basics of quantum field theory you should be fine. It is assumed you know what the stress-energy tensor is, how to make field variations etc. At the same time you should know what "normal" string theory is about, why we use Calabi-Yau manifolds (and thus all basics of the geometry associated to them). Then you need to know some representation theory in order to understand how the topological twists works to get the A and the B model and it should be in principle ok to get up to the free energy of topological strings.
Topological strings are a vast subject with incredibly rich mathematical structure. I would say that knowing basics of algebraic topology (for example cohomology) and basics of algebraic geometry (for example toric varieties, sheaf cohomology) is extremely useful and at some point unavoidable. It also depends a lot on what you want to do with topological strings. If you want to study moduli spaces you have to go even more formal and at least to me any connection with "proper" physics is in a grey area. To understand homological mirror symmetry, which is some sort of duality between the A and the B model, you need to understand category theory on top of other very abstract mathematical constructions. Or, you could be interested in the (refined) topological vertex and see how to compute topological partition function via the vertex and what is the relation to the corresponding SCFT's, i.e. the relation to a 5d or 4d Nekrasov partition function.
Vonk's and Collinuci's lectures (the first hits on Google) are very good places to begin learning some stuff. For a lot more info the yellow bible (not Di Francesco's CFT) of Hori et. al. "Mirror Symmetry" is pretty awesome. I warn you for a high level of mathematical abstractness (unless you are a mathematician which should be fine). There are some nice recorded videos of Ooguri as well here and here.