SPRAY -- no question. Unless you like turning it from a one or two day job into a couple weeks. And with lap board, you can shoot it from an angle to get the different angled surfaces in one pass, while rolling won't do it and brushing that sort of thing is very time consuming. Changing colors means you need to get good coverage, and those angled surfaces will really show up your "holidays".
I used to paint and finish drywall professionally, and spray work (industrial, high work and house -- interior and exterior) was my forte (airless for most, conventional for a lot of cabinets and finer stuff). I had a system with an airless for exterior house painting that works like a charm -- one day on almost any house (after prep) for me and one helper.
Surface prep work can be the most time consuming part, whether you spray, roll or brush. Once that's done, including any required priming (if large areas require priming, you can spray primer as first stage after masking), I'd set one day to paint the house, excepting trim color, which may or may not be doable in the same day, depending on house size and amount of trim.
Spray day, we start out as a 2 man team, masking windows, doors, etc. with 4 mil visqueen, a hammer-tacker (stapler) and cardboard drywall shims (they come 48" long, about 1.5" wide and about 1/8" thick)*. You take a piece of visqueen a little larger than the door or window you are masking, hold it against the top trim with a piece of shim and staple it with the hammer tacker through both into the trim, then go around the window the same way and trim with a utility knife. (If you don't use the shims, the hammer tacker can mark the trim, but if you do, each staple leaves only 2 small holes that will be invisible after painting the trim.) Masking tape and either visqueen pieces or masking paper are used for hose bibs and other small stuff that can't be removed. There is usually not too much masking tape type masking on an exterior.
I'd usually go part way around the house with the helper before I'd start spraying (a little slower because I'd also be doing his helper work moving stuff and rolling/brushing as necessary), but you may want to finish masking first. Your helper is going to speed your spraying by helping to move the ladder and drop clothes and using a roller or brush to catch runs, even out spots or cut in some spots from the wet paint you just sprayed on. Main trick for fast spraying: your helper (or you) has a slightly paint wet brush and roller ready to go to roll out or brush out any runs. You just get enough paint to cover on as smoothly as you can, and catch any runs before they begin to set up.** Use drop clothes and runners as you go 'round the house to keep it off decks, etc. You can just flat rip this way, and get full, even coverage on everything. (There are a few finesse tips if you have discoloring AND are using a semi transparent stain, but it doesn't sound like that's applicable here.) Also, remember to have a spray shield on hand -- a 4 x 2 foot piece of cardboard will suffice, which you either one hand, or your helper holds to allow you to get a spot where you don't want to spray an adjacent area (e.g., useful around eves, to shield plants, etc.).
When done spraying, peel the masking off -- a snap with the shims. And now you're ready to paint trim with a brush. BTW, real painters don't wipe the paint off their brushes after dipping them into the bucket. They tap both sides against the inside of the bucket to keep the paint from dripping while not taking all the paint off the brush. Up to a point, the more paint you can keep on the brush, the better it flows into corners, draws straight lines and gets the painting done.
* If you can't find drywall shims, you can mask the same way without them, but I'd recommend an electric or hand stapler instead of the much faster hammer tacker, since the hammer tacker is gonna leave marks. And you're gonna spend a little more time removing staples without the drywall shims.
** On interior stuff that is really flat, like finished sheetrock, esp. in new construction or industrial LARGE wall applications, any good painter is going to try to use an airless with a helper double rolling behind. It gets enough paint on to avoid holidays, even if it's all white and hard to see light spots after hours of staring at the same white and shadows. The airless gets the paint up there, the roller (or brush) only smooths it and evens it for a smooth, even and professional result.