Where to begin in approaching Stochastic Calculus? I have experience in Abstract algebra (up to Galois theory), Real Analysis(baby Rudin except for the measure integral) and probability theory up to Brownian motion(non-rigorous treatment). Is there a suggested direction I can take in order to begin studying stochastic calculus and stochastic differential equations?
 A: In my experience it is best to get to your goal as quickly as possible to maintain momentum.  You could read Royden and then Billingsley and finally start on stochastic calculus.  But personally, I would probably run out of steam before long.
I would recommend you read
Jeff Rosenthal's book A First Look at Rigorous Probability.
It is 200 pages long.  It is very clearly explained (baby Rudin is all you need).  It develops all the measure theory you need in a probability context.  It has a lot of easy exercises that build confidence that you understand basic concepts.  Half of these have solutions!
The last chapter whets your appetite for stochastic calculus and he gives suggested reading.
A: As previous answers have indicated, a solid measure-theoretic approach to probability is essential.
A book I would strongly recommend for the measure-theoretic approach to probability is:
R. M. Dudley - Real Analysis and Probability
I found it very clear, well-organised and invaluable when I was working on such things.
A: I would suggest to first take a course and/or get a book on probability theory, in view of measure theory. It helps to have studied measure theory first but is not necessary.
Some textbook ideas, but in no way an exhaustive list:


*

*Durrett: Probability: Theory and Examples. I've followed a class using this book, you can find an earlier edition online I think

*Billingsley: Probability and Measure

*Gut: Probability: A graduate course
I have not worked with the last 3 but only consulted for references. Perhaps our more experienced user could have a better say in the matter.
Once you have done that, you can take a class on stochastic calculus in general. That should explore the construction of Brownian motion, the Ito integral, some Stochastic Differential equations and a continuation of martingales that you will have started in  course 1.
Some books are


*

*Shreve, and also Steele have books with some financial emphasis

*Karatzas and Shreve's Brownian Motion and Stochastic Calculus has been around a while but might be harsh for a first class


You can then take more advanced class on specific topic such as Stochastic Differential equations. One book that comes to mind is Oksendal's
A: There are several textbooks out there for finance professionals that do not require heavy understanding of measure theory (don't require any).  I think there is an unconscious bias of math professionals to want to learn the theory.  But it's totally not required.  Just as an engineer can learn to solve integers and solve them well with lots of tricks, never having done theoretical calculus, so can you very easily learn the major methods of stochastic calculus, with no more prereqs than what you already have.
E.g. see https://people.emich.edu/ocalin/Teaching_files/D18N.pdf
BTW, nothing stops you from going back later and learning a bunch of rigor theory stuff.  But why not go now and get the basic manipulational skills.  They will do you more good in any sort of finance job.  And it preserves option value (decide to invest in the rigor later when you know you need it.)  
Actually I would even go further and say that you are more likely to succeed in learning the material, to learn it deeply, and even time efficiently, if you do it the "easy way" first and move only to rigor as a progression.
A: I Suggest 
For Measure Theory


*

*Real Analysis -Royden 

*Measure Theory- Halmos.


For probability theory, Brownian motion and stochastic Calculus


*

*"Probability with Martingales" by David Williams.

*"An Introduction to Probability Theory and Its Applications 1-2" William Feller.

*"Diffusions, Markov Processes and Martingales:1-2" by Chris Rogers and  David Williams.

*"Introduction to Stochastic Integration" by  K. L. Chung, R.J. Williams

*"Stochastic Differential Equations: An Introduction with Applications" by Bernt Øksendal.


You may also need to learn some Complex Analysis. Although Complex analysis is not essential to learn probability theory and stochastic processes. However, contour integration and Fourier transforms are indispensable tools and it is also one of the most beautiful and useful areas of mathematics. 
