My thinking is to ultimately get into a graduate program at a decent
school; but before I do that, I'll need to have a foundation in math
and science, and possibly a second degree in engineering or some
You are blowing the requirements a bit out of proportion. If you want to be a bonafide engineer, what you really need is a B.E.: an engineering diploma accredited by ABET or another respected accreditation agency. A B.E. generally takes about a year or a year and a half, not counting the prerequisite material. Generally speaking, the pre-reqs aren't that bad, if you are a naturally mathematically inclined person and can work hard: you'll need about 2 terms of multivariable calculus, 2 terms or so of college level physics, maybe a term of chemistry, and probably a computer science class. You can complete all of this and the B.E. in 2.5 years or less if you work hard - I know a number of people who have done it.
Let's say you're dead set on graduate school, but you want to get there as quickly as possible. You realize that once you're in grad school you'll be there a while, but you don't want to wait a long time to get there. I recommend the following:
- Apply to the B.E. program at an accredited college. For expediency, you won't want a liberal arts college. Maybe a decent state college with a fair sticker price and a staff willing to work with you to get you in and out as quickly as possible. You are not applying to do an A.B. or a B.S., just the B.E.
- Do you want a Ph.D. or a Master's? There are many 5-year B.E.-Master's programs out there, such as this one, this, this other one, and this, and also this, among many others. If you want a Ph.D., then do the B.E. and then apply for a Ph.D. program somewhere. I'd bet they'd appreciate that you are more adult, and experienced.
One last thing. When you think about what you want to do in life, it's important to think about what you want to do, as opposed to what you want to be. Do you want to be an engineer, or do engineering?
I am a mathematics major, and I think my future occupation(s) will be what most people would call "engineering." In my case, an undergraduate education in mathematics is, I think, more useful for the "engineering" which interests me than an orthodox engineering diploma would be. The trodden path is not always the best one, and especially when forced into unconventional positions as you are, thinking outside the box is a good idea.
I'd definitely encourage you to reflect on whether or not spending 1.5-2.5 years on a B.E. and another 1-3 on a Master's or 3-6 on a Ph.D. is necessarily the path to a job and life where you will be able to do the sort of engineering you want to do. You can always find awesome learning material for free online. A good diploma will do 2 major things for you: prove that you know a decent amount about something hard to learn, and teach you a decent amount about something hard to learn. Once you have those two things, you can always learn hard things on your own, and prove to anyone who asks that you do indeed know how to learn hard things without help.
EDIT: Apparently the "BE" program, or at least the BE nomenclature, is not as common as I had been led to believe. That said, in much the same way that there are licensure requirements for "pre med", there are licensure requirements for meeting the criteria of a Professional Engineer in the US. I think that if you can figure out what those requirements are (for example by asking any engineering department), you could probably take those specific classes but not complete an entire bachelor's and then apply to graduate school. That said, I'm not in graduate school yet (I'm an undergrad), so I'm not sure. I know people who have taken that type of route, but it may be a result of an unusual system at my college.