Have a look at Bump Automorphic representations. The adelic picture is given there as does Tate's Thesis. I think Bump works in the global field context there. The main difference is that the adelic norm map has discrete image in the function field case, but that doesn't harm much.

Although Tate's proof works as well in characteristic p, he assumes the number field setting right from the start. It is a simple exercise to extent everything to global fields and Tate actually refers to Riemann Roch as an equivariant version of Poisson summation in his thesis. Certainly, also Weil's basic number theory is worth a look.