Use a medium weight fusible interfacing to stabilize the t-shirt fabric. Take some of your own old t-shirts and try different interfacing weights to see which one works best. Some of them are too light and stretchy and don't work well. Some are too heavy and make the fabric too stiff.
Cut the t-shirts to separate the back from the front. Cut the interfacing at least an inch or two bigger than the size you want. Then, once it is ironed on, you can flip the t-shirt over and trim down to the desired size with the pattern from the shirt centered correctly.
The best t-shirt quilt I saw had each t-shirt with about 1.5 inches of black framing it and then another fabric between.
When you iron, iron from the back side. Some of the shirts have that plasticy stuff on them that melts (or maybe this is just kids' t-shirts).
Hope this advice helps.