Kline's book is not rigorous in the sense that it avoids the formal definitions and proofs that were developed so that calculus could be put on a "firm footing," at least until the last chapter. However, based on my experience, most calculus courses that use "standard" textbooks also don't emphasize these formalisms/don't do a good job of teaching them. As Tegh said, you would only be expected to know the "rigorous stuff" after learning real analysis, and many scientists never take a real analysis course.
See also this Reddit comment about the book.
Kline's book covers single-variable calculus pretty thoroughly. It does cover a decent chunk of multivariable calculus, but you would probably still want to supplement this with another text. For example, Kline doesn't cover vector calculus, which is important in many science fields.
After Kline's book, if you want to jump straight into dynamical systems, Steven Strogatz's Nonlinear Dynamics and Chaos is aimed at students that have completed a first course in calculus. Video lectures are available.