It sounds like you have a domestic sewing machine (DSM), so I would definitely recommend starting by stitching around each block to secure that area.
I have a longarm, and I have quilted several T-shirt quilts for one of my customers, usually with a pantograph that has a "theme" that matches the "theme" of the t-shirts. I did not have any problems at all stitching through the different logos, decals, whatever was on the t-shirt. I've done football themes (shirts were all from football camps, etc), western themes (shirts were horses, cowboy boots, etc), and I just did a feather panto on one quilt that had all kinds of t-shirts.
If you're on a DSM, I would recommend changing your sewing machine needle to a "JEANS" needle, to ensure that it goes easily through anything printed on the t-shirt. This should take care of the "rubbery" sections you mentioned. A large, all-over meander will serve you well for the stitching within the blocks of the t-shirts.