If you are buying everything from scratch, $1600 would be pushing it. A lock will be almost $100 if you buy a good one, another $50 for the helmet, maybe $100 for panniers, $100 plus for the sort of rain jacket you can ride in without feeling that it's raining inside when you ride. If you buy stuff that's too cheap for a long tour like that, it's more likely to break/tear/leak and all that. You might want rain pants, bike shorts, stiff mountain bike shoes....

I'd start buying bit by bit, and try things out with a few weekend trips before you even think about touring.

Another resource is www.crazyguyonabike.com, for all the journals of people who have done that sort of trip before.