A good basic plan, I'd recommend understanding the situation in non-wiggly coordinates first, the way it is presented in SR. This implies, though, not too much focus on acceleration, because the natural coordinate system of an accelerated observer is "wiggly". These coordinates not really wiggly in the spatial dimensions, but rather "wiggle" the time dimension. To thouroghly justify that statement I'd have to talk about the metric, which is one of the things I think you're trying to avoid at this point, so I won't clarify the vague general statement and hope it enlightens more than it confuses.Thanks, I think I can follow that chapter, but it will take some work. The first chapter is a little over my head, the notation is unfamiliar, although I can guess what it means.
But before digesting foundations for GR, I'd like to get SR straight.
What you can do with the SR approach easily is to describe the coordinates of an accelerating object in an inertial frame, and calculate some basic coordinate independent observations such as proper time and proper acceleration on the accelerated clock, maybe even calculate the proper time of emission and reception of a few radar signals. What you'll be lacking is the "viewpoint" of the accelerated observer, the most direct route to that viewpoint is to learn about metrics and some of the (non-tensor) basics of how they can be used to deal with "wiggly" coordinate systems, but that can wait until after you learn about the non-wiggly coordinate approach in SR.