Grounded -- First, please accept my apologies for being snotty and arrogant(the physicist's disease) I misjudged your zeal and passion for understanding. (Note, however, as my dear Mother told me -- You learn more with your mouth shut than with it open)
Physics is very difficult, for two primary reasons: it requires a real facility with and understanding of a lot of advanced math; the conceptual basis is highly abstract and idealized, in E&M, QM, SR, GR, ..., and is not fully consistent. So, particularly, for the last reason, most physicists are highly pragmatic -- do the best you can with what you have, don't sweat the hard stuff unless it messes up your work. As I'm sure you know, for example, there is huge controversy about the proper interpretation of QM, but many working physicists pay little attention to the controversy. The standard ways work just fine in practice.
Everyone who teaches physics, say SR, will tell you we did not understand SR until we taught it. It takes a long time, and many exposures to SR to get it right. And, most students of SR will tell you that sometimes you have to suspend your judgement to get to a good understanding -- that's the advantage of a good teacher, he/she can suggest what to sweat now, and what to defer. All told, SR is a huge subject, of which the basic space-time kinematics, the x's and t's, comprise only a very small portion. It is the totality of SR that is so compelling.
That being said, the path to a robust understanding of SR necessarily involves lot's of math -- there's no way around that fact.
Tom's explanation of the imaginary exponential is right on. The way, typically, you build up a comfortable intuition about i, the square root of -1, is to work with it, work with it a lot. It's part of the language of physics, and is used so much because it is a powerful tool in working with waves, periodic phenomena, much of QM, and many other branches of physics and engineering.
If you are serious about physics, you need to hit the books, and do the homework. Resnick and Halliday is a good place to start. And, absolutely necessary, learn calculus. Then you will understand that the x and t in the exponential are not x2-x1, t2-t1. That exponential gives you the compex value of the wave amplitude at x at time t.
Once a professor, always a professor.
Regards and good luck
Reilly Atkinson