So, what band developed the printing of their name on shirts?  What
was the first shirt they printed up?  Any thoughts?

Dear agruhn-ga,


The Beatles were certainly one of the first bands (but probably not
THE first) to print their name (and faces) on shirts. According to a
Dutch web site


"If one [...] believe[s] the April issue of the periodical Wiener, the
advance of the music t-shirt began with the Beatles. Beatles manager
Brian Epstein commissioned a textile printing company in Hamburg to
make more than 12,000 t-shirts with the four page-boy heads from the
Beatle's Star club performance of 1964."
http://www.mediamatic.net/cwolk/view/3157


You can find a picture of the original 1964 T-shirt at
http://www.fab4collectibles.com/SHIRTwithTAG.jpg


I'm not an official Google Researcher. I hope this answer might help you a little.
Xituculumucumba


Google search strategy:
"music t-shirt" sixties
://www.google.com/search?hl=en&ie=UTF-8&q=%22music+t-shirt%22+sixties