B. Øksendal, Stochastic Differential Equations, sixth edition, Springer 2007
Skips a lot of basic proofs. And the motivation and intuitions behind the models are not very clear, though its use of symbols are pretty precise. Though this course uses this book, but I found it too hard to follow as a beginner with only knowledge of real analysis.
Steven E. Shreve Stochastic Calculus for Finance II: Continuous-Time Models
Better than the above book, it has very simple examples to explain intuitions and motivations. The proofs are just in the right amount, it gives sketches of proofs and the connections between concepts and theorems are logically tight. (Highly recommended).