Well, what happened for me is that I ended up learning "vector calculus" at least three times: first in introductory calculus, next in analysis, and finally in differential geometry. Each pass through the material brought a different perspective and deeper understanding. So I'd say that the best book for you to read at this point really depends on what you've had before. Are you completely new to the subject? Have you seen the basic theorems but without proofs? Are you reasonably comfortable with the theory of grad, div, and curl and want to understand how it generalizes and relates to other branches of mathematics?
There are lots of good books on the subject--like many other students, I first learned the basics from Stewart's "Early Transcendentals" book. Later I had a substantially more rigorous course (analysis) using "Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach," by Hubbard & Hubbard, which is really a great book in my opinion--plenty rigorous, but also filled with various interesting odds and ends (for example, the general theory of Newton's method and superconvergence) that don't usually make it into texts covering this subject at the intermediate level. It's also somewhat idiosyncratic in notation and expository style--the authors tend to treat things very "concretely." If you're looking for a more abstract treatment, I'd recommend picking up a book on differential geometry, although if you haven't seen any vector calculus it's going to be a tough read. Vector calculus is really a subfield of differential geometry, but usually when the term "vector calculus" is used, it refers to a certain "lowbrow," 19th-century perspective on the material that's very physics-oriented; by contrast, most differential geometry authors assume familiarity with this approach and generalize from there. Just to clarify, though, it's possible to be completely rigorous within the framework of "classical" vector calculus; differential geometry is just a different, more general perspective on the same material.
However, even if you're looking for the most mathematical rigorous treatment you can find, no matter what stage you're at, I'd really recommend supplementing whatever you read with some more intuitive, "physics-style" exposition--after all, the subject was invented to accommodate the needs of theoretical physics, and I think you'll find that having a firm grasp on the physical meaning of all the abstract math involved really helps your overall comprehension. Studying physics in depth immensely improved my mathematical comprehension, even though physicists themselves tend to be fairly non-rigorous. Lots of physicists swear by "Grad, Div, Curl, and all That," by H. M. Schey. Also, any physics textbook on electromagnetism or classical mechanics (try Griffiths or Jackson for E&M and Goldstein for mechanics) should cover basic vector calculus; if you're interested in learning a little differential geometry, general relativity textbooks are good places to look for intuition (I recommend Carroll as an excellent introduction; Misner, Thorne, & Wheeler's book is longer and more idiosyncratic, and Wald's book is extremely rigorous but quite tough to get through).