Hi Susan and welcome to the board!
I am wrapping up the charmed quilt I made from my daughter's baby clothes. Moonpi is exactly right you need interfacing. At least medium weight, all you need to do it affix the interfacing before you cut
A note before beginning, I found that even though my daughter's clothing had be stored away in plastic bags the fibers contained things such as dust that would make my eyes water and my nose run, if I had to do again I would WASH it all before cutting.
You would start by cutting out the seams, don't worry unless it's something small you wont get enough from the seams to make it worth seam ripping. I know it's hard to cut up those things but take a deep breath, turn your first shirt inside out and cut down the side seams, if you line it up with your fingers you can cut out both sides of the seam at once. When you cut out the seams you can also cut off the buttons to save for later to make a button box.
After cutting out the seams roughly fold the shirts, so there will be less to iron later. I used a woven basket to hold all my "cut clothes". By the time you are done cutting you need to have determined what sort of layout you would like to have. I'm sure you can fine plenty of ideas by searching for "shirt quilt". I used a pattern of my own making which you are more than able to do as well. It's really all about how you want it to look when you're done. There are "crazy quilts" where you cut random strips sew them together lengthwise and then cut them crosswise to make random shapes and sew them all back together, no mistakes that way, and of course you can cut "blocks", just be sure to look at the sizes you have, things like ties will make smaller "squares" than a shirt.
Now that you have your pile of cut clothes you would use the interfacing of your choice on the knit fabric, as in you pull on it different ways and it gives more than 1/8". some denim, Tshirts, terry cloth, fleece, ect. all need to have interfacing to prevent it from stretching during handling and while you work. A note about interfacing, it should stop the "give" in the fabric, if the piece still can be pulled out of "shape" after you've interfaced it, you need a heavier weight. I needed much much more than I ever dreamed on, so don't worry if what you get the first time isn't quite right. (that's of course baby clothes and you may find less of your shirts need to be interfaced)
After you have interfaced your fabrics, then you can use a rotary cutter and mat to cut the clothes to the size you need for your layout/pattern. It sounds like a pain, but really it's not, it's just a bit of work. And of course there is always help just a post or pm away. :D
You will be so glad you decided to do this.