I have sewn on many patches (scouts, karate, you name it). A sewing machine can be useful but in the end I found I preferred the time honored method. Doing it by hand using a heavy duty needle, sturdy thread (tell the people at the sewing store what you are looking for, this is not a WalMart type thread anymore), and a pair of needle-nose pliers. My husband used to sew some on as well. The good old "stab and grab" method worked the best. You can really adjust the size of stitches that you need doing that.
Just as an after thought. Did you realize boys outgrow those scout shirts really fast? Don't make your stitches too small. You're going to be moving them to another shirt. Ah...memories...