In talking about primers, the first thing to consider is what you want the primer to accomplish.
In the world of painting in general, primers do several things. First, they seal the surface of the base material, so the finish coat has a smooth surface and nasty stuff growing or otherwise developing in the base material doesn't come out through the paint. Second, they establish a physical and/or chemical bond with the base material, making the finish coat of paint stick better. (In some instances the finish coat wouldn't stick at all in the absence of a primer.) Third, they provide a uniform color, making it easier for the finish coat to cover the color of the base material.
In plastic modeling, the only one of those functions that's relevant is the third one. Modern styrene plastic doesn't have any pores to fill, and, so far as I know, doesn't leech or otherwise secrete any nasty substances. (In the Goode Olde Dayes, modelers frequently carved hulls from pine, and coated them with shellac or some other primer to keep the pine juices from working their way out through the paint. That isn't necessary on plastic.) Some people seem to think that primer makes paint stick better to plastic, but, except in unusual specific cases, that just isn't true. Modern hobby paints, be they acrylic or solvent-based, are specifically formulated to adhere to styrene. Putting a coat of grey or red enamel on a piece of plastic is not going to make a white finish coat stick any better. (Metal is another matter. Metal primers do make hobby paints stick better to metal parts.) Washing the parts in advance isn't a bad idea; it should get rid of any residual mold release. (I confess I've never had a problem with mold release, but other modelers say they've had difficulties with it.) If the parts are clean, a decent, properly stirred hobby paint should have no trouble adhering to them.
Modern hobby paints are excellent; in most cases they're perfectly capable of covering styrene of any color in one coat. That's especially true if they're airbrushed. Brush painting a large area of white on a hull that's molded in black is more difficult, and a primer (light grey, maybe) probably will help. It's also true that, though a single finish coat may look opaque, it may not be in reality. If, for instance, parts of the model are molded in yellow and others are molded in black, and you paint all of them white, the white may look a little different on the yellow parts than it does on the black one. The same logic applies if you use fillers (e.g., grey or white putty on black plastic). The outlines of the filled areas may show faintly through the finish coat. You then have two options: apply additional finish coats till the difference disappears, or use a primer.
I try to avoid primers on plastic when I can, simply because every additional layer of paint, by definition, obscures the underlying detail at least a little bit. If I were painting a white bottom on a hull molded in black, though, I'd probably spray it with a light grey primer first. My personal preference is to use solvent-based paint (e.g., Testor's Model Master enamel) for the primer and acrylic (e.g., PolyScale) for the finished coat. (I'm a big fan of modern acrylic hobby paints.) The enamel dries fast, sticks firmly, and doesn't have any tendency to soften up when the acrylic finish coat is applied. Some modelers do it the other way around: they start with an acrylic base coat and add washes and other finish techniques using enamels. I'd be a little concerned that the solvent-based paint on top would soften up the acrylic underneath, but if the photos in FSM can be believed, some modelers have gotten excellent results that way.
One other thing ought to be born in mind. Acrylic and enamel paints behave differently when they dry. I'm not a chemist or a physicist, but in layman's terms (the only ones I understand), enamel dries from the inside out and acrylic dries from the outside in. Acrylic feels dry to the touch almost immediately, but takes some time to "grab" the surface. (Just ask anybody who's painted the woodwork in his house with latex paint, which behaves in much the same way.) In the early days of acrylic hobby paints, lots of modelers got frustrated with them because they rubbed off so easily. In fact, though it may take a couple of days for the reaction to take place, acrylic will "grab" styrene pretty firmly - as firmly as enamel, or nearly so. Modern enamels dry much faster than the old stuff did - and when enamel is dry on the surface it's safe to assume that it's adhered pretty firmly to the base material. That's one reason why, for me personally, it makes sense to use enamel as a primer under acrylic.
The sequence for assembling and painting the hull components is, of course, up to the individual modeler. Generally it's best to leave the painting until as late as possible - that is, until you've assembled as many parts as you can without their getting in the way of the painting process. (Obviously you just about have to paint the lower deck before you add the one above it.) Another consideration is the fit of the parts. That's an old kit. I built mine a long time ago and I don't recall any major fit problems, but folks who've bought Heller sailing ships recently report some pretty serious problems with warpage, shrinkage, etc. If you're going to have to use any fillers on the hull components, the time to paint the hull obviously is after the filler is in place.
Hope that helps a little. Good luck.