Professional actors rely on "dialogue coaches". Have you ever viewed "The King's Speech"? That seems like a parallel experience.
I haven't deliberately picked up new English accents, but I've taken formal classes in Spanish, making conscious efforts to imitate the phonemes of native speakers.
I would say that consistency will count for a lot in choosing your dialogue coach. Every year I had a new teacher; the first was Mexican, the second was American, and the third was from Barcelona! So, I was faced with deciding which accent(s) to learn and how to distinguish them and how to avoid mixing them all together. So, identify your target culture, and articulate that to your prospective coach so that you're on the same page, and try not to change coaches once you've settled into a relationship.
So you'll soon find out that there is not one "West Coast American accent", although there may be fewer of those than British English has produced. Growing up, I learned how to communicate with Valley Girls, surfers, CEOs, computer geeks, vatos, Crips, Bloods, goths, rivet-heads, Christians, pagans... well, you get the picture!
And before anyone objects and says those aren't accents, "surprising a native speaker" will involve vocabulary, grammar and syntax as much as the phonemes and inflection. So the OP has a broad definition of "accent" in mind.