If you're getting the shirts screen printed, the maximum width and height are dependent on the size of the screens and platens used by the print shop. So there really isn't one answer here. If you know what print shop you intend to use, ask them what their max print size it. If you just want a general answer, pull a t-shirt out of your closet and measure out how you want the design to look on the shirt. Then when you choose a printer, you can revise the design to fit within that area. Generally, 18" x 20" would be considered an over size print.
Adobe Illustrator is used to create vector artwork, which is pretty common for t-shirt designs.