I brush the whole thing [people don't usually break out a sprayer for
one door]. What I do is paint the panels first - well, after painting
any edges that wrap around. I keep a wet rag handy so I can wipe the
edges. Then I do the stiles and rails, following the grain like on a
wood door.

...with a small sponge roller and a brush for around the panels. paint
it on a hot day.

oh yeah, that's the best idea ever.

Anyway, I don't like seeing roller texture on a paneled door,
especially at the front entrance. just doesn't look as good as it
could. And unless your a big fan of the glossy look, I'd suggest using
a satin finish latex. Be sure to get some latex paint additive - it'll
slow the dry time enough that you'll appreciate it.

Also, if you're considering a red color, avoid anything that is an
ultra deep base [unless you want to do 3+ coats]. I know Porter and
Sherwin-Williams [and others, I'm sure] have a red base. I've used it a
few times and it's 1 or 2 coats, max. Since you're going over white I
would count on 2 coats.

Do as much as you can before the install.