I don't have my copy to look at now... in particular, the entire quote. It may be that "non-mathematical" means "non-tensorial" for Zapolsky... I'm not sure. However, for textbooks, Spacetime Physics is great at introducing and emphasizing more of the physics. Certainly, there is mathematics that must be used to obtain numerical and algebraic results.
In my opinion, you can [begin to] appreciate some of the content with your current preparation (being in College trig and some physics in high school). Some basic physics you'll need for that text: Kinematics, Force and Newton's Laws, Work and Impulse, and the Conservation of Energy and of Momentum.
With the worked problems, you may be motivated to brush up or learn some more mathematics.
You may wish to get help by posting HW-type questions in the Homework Forums. For conceptual questions, you can post in the Relativity Forums.
If you want something less mathematical [but actually far deeper conceptually], try
Geroch's "General Relativity from A to B"
If you want something comparable to Spacetime Physics, but more modern, try
Moore's "A Traveler's Guide to Spacetime"
Frankly, I'd stay away from the "pop-science" books to learn relativity... but they are useful for motivation to find a good book to learn the subject.
Good luck.