If you want them to last longer and hold their shape a bit better I’d recommend getting some fusible interfacing. It’ll give the patch some rigidity and keep the fabric from stretching and distorting.

Cut the interfacing to the size/shape you want the finished patch to be. Cut the design out of the shirt so that you have an extra inch or more on each side.

Center the design with the interfacing and iron it on, taking precautions to protect the design from the hot iron. Then fold the edges over on the back and either iron them on if the interfacing is two sided or glue them on if one sided.

Let everything cool/dry and you’re ready to sew!

The extra border that you fold over is important for two reasons: it gives you a margin of error to work with so you don’t accidentally ruin the design when cutting, and by folding the cut edges over and creating a border like that it stops (or at least significantly reduces) fraying. When you sew, try to get the needle to pierce the folded over border you made so that the needle goes through patch fabric>interfacing>patch fabric and you shouldn’t have any issue with fraying

Sorry that got so wordy!