What are the prerequisites for stochastic calculus?

Stochastic calculus relies heavily on martingales and measure theory, so you should definitely have a basic knowledge of that before learning stochastic calculus. Basic analysis also figures prominently, both in stochastic calculus itself (where limit procedures of various kinds appear, as well as the occasional Hilbert or $L^p$ space argument) and in martingale theory itself.

Summing up, it would be beneficial for you to first familiarize yourself with elementary mathematical tools such as:

-Real analysis (e. g., Carothers "Real analysis" or Rudin's "Real and complex analysis")

-Measure theory (e. g. Dudley's "Real analysis and probability", or Ash and Doleans-Dade's "Probability and measure theroy")

and furthermore learn basic probability theory such as

-Discrete-time martingale theory

-Theories of convergence of stochastic processes

-Theory of continuous-time stochastic processes, Brownian motion in particular

This is all covered in volume one of Rogers and Williams' "Diffusions, Markov processes and martingales", and also sporadically in the two probability-related books above by Dudley and Ash and Doleans-Dade.

With a background like that, you should be well prepared to learn stochastic calculus, which you can do from volume two of Rogers and Williams' "Diffusions, Markov processes and martingales", or Karatzas and Shreve's "Brownian motion and stochastic calculus".

To gain a working knowledge of stochastic calculus, you don't need all that functional analysis/ measure theory. What you need is a good foundation in probability, an understanding of stochastic processes (basic ones [markov chains, queues, renewals], what they are, what they look like, applications, markov properties), calculus 2-3 (Taylor expansions are the key) and basic differential equations.

Some people here are trying to scare you away. It is certainly possible to apply stochastic calculus and gain an intuitive understanding of what's going on without knowing the details of a mean square limit or how to prove a function is square integrable in Lp space.

After all, it is a tool that first came into being for thermodynamic processes. Many of the Physicists who use it don't concern themselves with martingales and measure theory and all that in their applications.

Don't get me wrong, Functional analysis/ pdes/ measure theory are all interesting topics, but they are not necessary.

I'd recommend the book: "Paul Wilmott Introduces Quantitative Finance"