I want to stress that there is a huge difference between engineering math and math as mathematicians know it. I'll define math as "the thing that mathematicians do" and assume that is what you are trying to learn. I don't mean to insult you, but the background you described means that you probably know very little math, if any.

That being said, it's never too late to start! Contrary to what others have been recommending, I do not recommend starting out with baby Rudin, especially if you have never written a proof before. I would say your best bets are as follows:
Absolute Beginner
By this I mean, you have little, if any, experience writing proofs. I do not recommend many of those "learn proof techniques" books because they often make math seem like a bag of ad hoc tricks. I would recommend:
Basic Mathematics by Serge Lang (Basic Mathematics: Serge Lang: 9780387967875: Amazon.com: Books)
Naive Set Theory by Halmos (Naive Set Theory (Undergraduate Texts in Mathematics) 1974 Edition by Halmos, P. R. [1998]: Amazon.com: Books)
Geometry Revisited by Coxeter (Geometry Revisited (Mathematical Association of America Textbooks): H. S. M. Coxeter, Samuel L. Greitzer: 9780883856192: Amazon.com: Books)
I think that these books provide a introduction to mathematical thinking "in context" and should help you decide if math really is for you.
Beginner
At this stage you have either done one or more of the books above. Maybe you have some experience writing proofs from an engineering mathematics class, or physics. At this stage you have a solid understanding of precalculus fundamentals. Contrary to what is usually taught in schools, calculus is not necessarily the only next step. I would recommend:
Calculus by Spivak (Calculus, 4th edition: Michael Spivak: 9780914098911: Amazon.com: Books)
Linear Algebra by Hoffman and Kunze (Amazon.com: Linear Algebra (2nd Edition) (9780135367971): Kenneth M Hoffman, Ray Kunze: Books)
Concrete Mathematics by Knuth (Concrete Mathematics: A Foundation for Computer Science (2nd Edition): Ronald L. Graham, Donald E. Knuth, Oren Patashnik: 0785342558029: Amazon.com: Books)
I think that the first two options here are absolutely fundamental, but it is a damn shame that a lot of math students never study discrete math. Also discrete math is fairly accessible without a lot of background, and you can find yourself working on interesting problems fairly quickly.
Core Undergraduate
This level would be for you if you have completed the beginner stage, or what is typical of an affiliated math major at a research university. There are three "core subjects" that math degrees universally require. They are:
Analysis
My favorite books are:
Principles of Mathematical Analysis by Rudin (Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics): Walter Rudin: 9780070542358: Amazon.com: Books)
Mathematical Analysis by Apostol (Mathematical Analysis, Second Edition: Tom M. Apostol: 9780201002881: Amazon.com: Books)
Calculus on Manifolds by Spivak (Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus: Michael Spivak: 9780805390216: Amazon.com: Books)
Algebra
Here I would recommend:
Abstract Algebra by Dummit and Foote (Amazon.com: Abstract Algebra, 3rd Edition (9780471433347): David S. Dummit, Richard M. Foote: Books)
Topics in Algebra by Herstein (Topics in Algebra, 2nd Edition: I. N. Herstein: 9780471010906: Amazon.com: Books)
Algebra by Artin (Algebra (2nd Edition): Michael Artin: 9780132413770: Amazon.com: Books)
Geometry/Topology
You should look at:
Topology by Munkres (Topology (2nd Edition): James Munkres: 9780131816299: Amazon.com: Books)
Differential Geometry by do Carmo (Differential Geometry of Curves and Surfaces: Manfredo P. Do Carmo: 9780132125895: Amazon.com: Books)
Non-euclidean Geometry by Coxeter (Non-Euclidean Geometry (Mathematical Association of America Textbooks): H. S. M. Coxeter: 9780883855225: Amazon.com: Books)
You don't have to grind through all of the above, each of those books has a wealth of wisdom. You should aim to do one or two from each group.
Other Undergraduate
This is where things start to really diverge. Honestly, with solid foundations in each of the three above areas you could start tackling the graduate level texts in those subjects. By this point, you should have enough "mathematical maturity" to pick up any undergraduate level book and get going with little resistance. For more variety, I'll give you my favorite books in a bunch of other areas that an undergrad might be interested in:
Differential Equations
Differential Equations by Birkhoff and Rota (Ordinary Differential Equations: Garrett Birkhoff, Gian-Carlo Rota: 9780471860037: Amazon.com: Books)
Partial Differential Equations by Strauss (Amazon.com: Partial Differential Equations: An Introduction (9780471548683): Walter A. Strauss: Books)
Probability
Basic Probability Theory by Ash (Basic Probability Theory (Dover Books on Mathematics): Robert B. Ash: 9780486466286: Amazon.com: Books)
Essentials of Stochastic Processes by Durrett (Amazon.com: Essentials of Stochastic Processes (Springer Texts in Statistics) (9781461436140): Richard Durrett: Books)
Number Theory
Analytic Number Theory by Apostol (Introduction to Analytic Number Theory (Undergraduate Texts in Mathematics): Tom M. Apostol: 9780387901633: Amazon.com: Books)
Introduction to the theory of Numbers by Hardy and Wright (Amazon.com: An Introduction to the Theory of Numbers (9780199219865): G. H. Hardy, Edward M. Wright, Andrew Wiles, Roger Heath-Brown, Joseph Silverman: Books)
Complex Analysis
Complex Analysis by Stein and Shakarchi (Amazon.com: Complex Analysis (Princeton Lectures in Analysis, No. 2) (9780691113852): Elias M. Stein, Rami Shakarchi: Books)
Complex Analysis by Ahlfors (Complex Analysis: Lars Ahlfors: 9780070006577: Amazon.com: Books)
Fourier Analysis
Fourier Analysis by Stein and Shakarchi (Amazon.com: Fourier Analysis: An Introduction (Princeton Lectures in Analysis) (9780691113845): Elias M. Stein, Rami Shakarchi: Books)
Combinatorics
A Course in Combinatorics by van Lint and Wilson (A Course in Combinatorics: J. H. van Lint, R. M. Wilson: 9780521006019: Amazon.com: Books)
Algorithms
Algorithm Design by Kleinberg and Tardos (Algorithm Design: Jon Kleinberg, Éva Tardos: 9780321295354: Amazon.com: Books)
Introduction to Algorithms by CLRS (
amazon.com
Introduction to Algorithms: Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein: 9780262033848: Amazon.com: Books)
The above should not be considered exhaustive. By this point, you are on the level of most math major upperclassmen.

After going through the above you'll be ready to start doing some real mathematics. Pick up a graduate level text, or a research paper in the field of your choice and get cracking. Have fun!

Edit** Added links and fixed some typos.