Grubb's recent Distributions And Operators is supposed to be quite good.
There's also the recommended reference work, Strichartz, R. (1994), A Guide to Distribution Theory and Fourier Transforms
The comprehensive treatise on the subject-although quite old now-is Gel'fand, I.M.; Shilov, G.E. (1966–1968), Generalized functions, 1–5,.
A very good,though quite advanced,source that's now available in Dover is Trèves, François (1967), Topological Vector Spaces, Distributions and Kernels That book is one of the classic texts on functional analysis and if you're an analyst or aspire to be,there's no reason not to have it now. But as I said,it's quite challenging.
That should be enough to get you started.And of course,if you read French,you really should go back and read Schwartz's original treatise.