I have made lots of t-shirt quilts and I like making them also. There are a few tips that you learn as you go along that does make it easier each time. I no longer bother to remove sleeves etc. before cutting my blocks, I just lay the shirt flat and use my Shape Cut ruler to cut around the logos.
I have found that using narrow strips of waxed paper laid over the rubbery or other difficult to sew over design images solves the problem of sticking or dragging of the sewing foot. I try not to have to sew thru the waxed paper but if I do have to, simply run the back of your seam ripper point over the seam and the paper tears out easily. Also, if you need to iron over this type image, all you need to do is use a piece of shirt scrap to lay between your iron and the image, i.e. a pressing cloth. The main thing is to not touch your iron directly to the image. This works well when you need to use the iron to secure the backing to the shirt square.
When you are sewing regular fabric to the t-shirt fabric, it works better if the t-shirt fabric is on the bottom and the woven (not knit) fabric on top. This way it will feed thru the machine more evenly.