I've looked at many expositions of relativity over the years, so I'm perhaps a little removed from learning for the first time. I'll try to help you - see whether you can benefit from the following comments.
For someone with precalculus, the most important thing to do is UNlearn something - lose all the cultural baggage we have about the nature of time and, in particular, you have to obliterate utterly any notion that there is one, universal time. This is hard to do, given Time's deification as an all-slaying Shiva in human culture ("This thing all things devours ....") and the best way to do this, I believe, is to go back to a wholly experimental conception of time. What can we say about it from the standpoint of rigorous experiment? Probably the main thing is the observation that unrelated physical processes under the same conditions tend always to run at the same relative rates. This simple observation of repeatability is what allows a sound notion of time in the first place. See my answer here.
Now, what happens to these relative rates if the compared physical processes take place in different inertial frames? Well, the relative rates change. Once we forget all the cultural time baggage, there are no surprises here. Change experimental parameters and a following change in the result are absolutely the norm, or at least extremely common, in the experimental world. So we need to learn about the rules that govern these changed relative rates: these rules are the Lorentz transformation. Crowell's book:
B. Crowell, "Special Reltivity"
gives a good exposition in the first chapter of the important experiments, such as the Hafele Keating experiment and the Rossi Hall experiment, as well as the latter's modern equivalent (measuring muon decay rates in accelerators). It also gives a thorough exposition of the Lorentz transformation and many exercises.
Another important thought experiment and corresponding postulate to be familiar with is Galileo's postulate that there is no experiment that can detect relative uniform motion from within one's own laboratory frame. One of the best recent expositions of this that I know of is Chapter 17 of:
R. Penrose, "Road to Reality"
Once you have done some exercises with the Lorentz transformation, which you can do with the exercises in Crowell's book, you may wish to further your understanding with a deeper, more "axiomatic" theoretical approach. The paper:
Palash B. Pal, "Nothing but Relativity," Eur.J.Phys.24:315-319,2003
uses only precalculus - mainly linear algebra - as does (almost)
Jean-Marc Levy-Leblond, "One more derivation of the Lorentz transformation"
and these two deal with the fascinating approach essentially due to Ignatowski. This approach begins with only very general symmetry assumptions about the universe and show that the experimentally observed dependence of relative physical process rates on motion frame has to be the Lorentz transformation. In particular, these approaches derive the existence of the speed parameter $c$ from these symmetry assumptions alone without any reference to light whatsoever. What they don't tell you is the value of $c$, only that it exists, governs the numbers that come out of the Lorentz transformation and has certain properties (e.g. that it is frame-invariant, i.e. witnessed to be the same for all inertial observers). Light, of course, was historically important for Einstein, because the speed of light was found, through the Michelson Morley experiment, to have this invariance property. So we see that $c$ is not primarily the speed of light at all - it just happens to be an experimental fact that light's speed is this special parameter, and this experimental fact can be taken to tell us something about light from special relativity, rather than the other way around, which is what Einstein did. That is, amongst other things, this experimental result tells us that light has to be mediated by a massless particle. For these reasons nowadays you often hear $c$ referred to as the "universal signal speed limit" or "universal invariant speed" or something of that ilk, rather than the speed of light. It just so happens that light's masslessless gives us a way to measure $c$ through measuring light.
These comments are not meant to be comprehensive, just to be an outline of an approach which may help your quest.
Something else you may be interested in is this video, "The Illusion of Time" by B. Greene, particularly beginning at 5:55. Although Brian Greene seems a little unfashionable on this site, and although he's not of course a real historian, I love this historical explanation: it casts a light on the late nineteenth century, and Einstein's relationship with that century's technology, in a light that I had not seen before. Greene shows how the rise of train networks, and the attendant need for precise synchronization, made us think, for the first time, deeply about time. Before then, synchronization was not important, and so we never really probed the experimental aspects of time deeply before trains. So our whole cultural conception of time was essentially grown from a badly uniformed standpoint. In this light, it seems very UNsurprising that we're surprised when we look at the experiments carefully. We don't see the subtle effects because $c$, relative to the speed that everyday things move at, is so big. Jerry Schirmer, another user on this site, makes the insightful observation (I can't find it, it was a comment) that this "bigness" really reflects the energy levels of nuclear as opposed to chemical processes. Our lives, hearts, the rate at which things play out, are governed by chemistry: energies in chunks of bond energy size (electron volts), whereas $c$ governs the energy and speed of nuclear processes. If life had been built on nuclear processes, everyday speeds to us would be comparable to $c$, and our "commonsense" conception of time would be very different. There is a science fiction novel to this effect, as I describe here.