A question from an engineering undergraduate

You're probably in your early 20s, so: relax, you're okay. There's plenty of time to learn math. You have not missed some sort of mandatory train for becoming a mathematician or even just doing math at a reasonably advanced level. College is a great time to start doing mathematics.

If you finished Spivak in just two months (depending on the detail you went into), then you're probably already a great deal more capable than most of the students actually doing the math major. So don't think that you have too long of a way to go; you're obviously able and interested, and between the two of those, you can probably complete some sort of math curriculum pretty quickly. I know that I was in that situation as well, at one point. I ended up knocking out the entire math major in a single year by taking all core sequences concurrently. Caution: your mileage may vary.

If you can (i.e. if it fits into your program in terms of scheduling), I'd definitely recommend doing a math major. If you're looking into going into engineering, you must know linear algebra, so take that. Multivariate calculus is also recommended. Beyond that, real and complex analysis are also staples of applied mathematics, along with courses in differential equations.

As an engineer interested in mathematics, you might want to look into the field of Continuum Thermomechanics. There are (applied) mathematics departments which offer such courses; yours might be such a school. Since you mentioned that you have done some self-study, books to look at as an introduction include:

1) The Mechanics and Thermodynamics of Continuua, Gurtin, Fried, & Anand

2) The Mechanics and Thermodynamics of Continuous Media, Silhavy

3) Many other freely available texts/sets of notes which are easily found online.