i studied/degreed in art, and later decided to pick up programming (sadly started with assembly, and now many years later working with C#).. so i think i can completely relate to the OP.
the odd thing about all of this, as a profession... i did NEITHER as a whole, i became a technical manager (the middle man of IT.. you know, sorta like the guy in this scene from "office space": [media]
[/media] ...)
does it hurt to understand how a part of your projects workflow functions to do what they do? nope. is it necessary to do it as well as they do? nope. but i think it can help to understand.
I honestly believe artists and programmers are very similar... creative, complex, and abstract thinkers.
so, to begin programming? I would say to NOT pick any particular language, but rather learn the concepts to build a good understanding.
Simon Allardice did a couple great training videos on Lynda.com that I took and found to be excellent starting places, "Foundations of Programming: Fundamentals, and Object-Oriented Design". They are mostly language independent, but show a few examples of concepts in different languages to show syntax differences, etc...
then i would say, pick a language that YOU want to learn.