I don't know what your experience level is in general (airplanes/tanks/cars how long, etc.) so this is aimed at a first-time ship builder who has some experience with other generas.
I tend to work up and out, so that means main hull, deck, superstructure. I assemble the superstructure main pieces and glue that to the deck before adding any details such as guns, reels, etc.. I prefer to do it this way so I can fill and sand any joints. There's a lot less of them and they're finer on Dragon kits, but the detail is so much finer it can be hard to do that without destroying it (whereas with most Trumpeter kits I'd just sand it all down and build it back finer anyway).
So, on Laffey I'd do hull, main deck, superstructure and funnels as one, unpainted mass. Then, paint from the bottom up. Even though the main color might be lighter than the hull red and black boot stripe, I find it easier with the hull curves to mask for red --> black --> topside and the "steps" in the paint aren't as visible as they're all "down" when looked at normally (some builders to topside --> hull red --> boot topping and I can usually see a "step" up to the boot topping from the top side paint). Laffey should be easy as she's essentially a single color on her vertical surfaces (as opposed to a dazzle scheme) - I find it easier to start with the vertical surfaces and then mask them off and paint the deck blue.
Parts won't stick as well once there's paint down, but the tiny parts are going to be fragile anyway so it's even odds in my book if this is really a problem. I tend to clean up the joints anyway if I can - the male side is generally fairly easy to scrape clean, and a sharp #11 blade can make it in to a lot of holes, but sometimes I'll dip the tip of a toothpick in Mr. Color thinner and then poke it into the female recess to let a small amount fill it and soften the paint enough to make the scraping easier.
I generally put the railings on last and touch up the paint. I use Gator Grip Glue as I find it easier to work with and believe it to be stronger over time than CA glue, which can get brittle.