I soak in oxy-clean while I'm scrubbing down other equipment, then spray it down, to get hop gunk off the outside. I don't worry about the inside at all; garden hose water isn't fit for any brewing purpose anyway, so why bother?
Warm water with a couple glugs of white vinegar will get rid of corrosion and leave the copper nice and shiny, but it's been a while since I've bothered with this step; the oxy-clean-and-rinse alone leaves it clean enough for me, at least so far.