A three-part blog post on how a theoretical physics paper gets made: inspiration, calculation, culmination. This tells the story through the example of a particular paper on cosmological inflation. (From Cocktail Party Physics.) The comments are probably worth reading, too.
Somewhat relatedly, although more about the mechanics of writing, Terence Tao on rapid prototyping of papers -- basically, sketch out the outline of the paper first, making the statements of the key intermediate results, and then fill in the gaps, rather than writing from beginning to end. I can't vouch for this working on the level of writing research papers for the simple reason that I have written none. (I hope this changes soon.) But it seems to work reasonably well for writing, say, solutions to rather involved homework problems that can take a few pages, and have three or four major intermediate results.
Also, Can Scientists be Great Communicators?, from The Accidental Scientist. I would say that regardless of whether or not we are (and I'm including mathematicians in this "we"), we have to be. This is true both in terms of communication among scientists (which is tremendously useful for driving along the whole scientific enterprise, because otherwise we'd all be reinventing the wheel) and in communication with the non-scientific public, which I think is quite important. For one thing, ultimately a lot of the money that funds science comes from taxes; if these people are in the end paying our salaries, don't we owe them some explanation what we're doing? But also, communicating complicated ideas in non-technical terms forces us to actually understand them. Feynman, when he was preparing his famous freshman physics lectures at Caltech, said that if he couldn't reduce something to the level where he could explain it to freshmen, it meant that he didn't really understand it. When you can't fall back on technical terms and convoluted equations you have to understand what you're doing. So communicating with the hypothetical "educated layman" perhaps pays dividends within science as well. I just wish that people didn't automatically glaze over when they heard I'm a mathematician, though...
Communicating with this person is becoming more and more feasible thanks to the web 2.0-ification of science. Write something. Google will find it. You'd be surprised to see how many hits I get from what looks like people trying to buy used furniture, for example. And although I offer no advice there on how much used furniture should cost, I feel like I'm doing something by just exposing them to the idea that perhaps mathematics can be used to figure out such things. It's a subtle propaganda campaign.
Another subtle propaganda campaign might be the sculptures at Bathsheba Sculpture (by Bathsheba Grossman), which are for the most part models of various mathematical objects, done via 3D printing in metal; she has both mathematical and artistic training. What other sorts of training might be useful for mathematicians?