Schwartz space of functions with values in a Frechet space

$\mathcal S(M,E) = \mathcal S(M)\bar\otimes E$ for the completed injective or projective tensor product, which agree since $\mathcal S(M)$ is a nuclear spaces. See H. Jarchow. Locally convex spaces. Teubner, Stuttgart, 1981. Fourier transform you can apply just to the left hand side of the tensor product.

Also, have a look at page 533 of the book of Treves you cited. This is treated there.

As an addendum to the above response, you might be interested in the fact that Schwartz wrote a sequel to his classic under the title "Théorie des distributions vectorielles" which gives an exhaustive treatment of this theme. It appeared in the Fourier Annals and is easily available online

Of course, the ultimate reference on this kind of stuff is the sequel mentioned by corserine, namely, this paper and this other one by Schwartz. I did not get a chance to look at the book from Peter's answer, but another useful reference (available online!) is "Vector-Valued Distributions And Fourier Multipliers" by Herbert Amann.