If your existing millwork on the inside is painted a glossy white enamel that everyone thinks is so cool right now (not me) then the interior paint part gets a little trickier. You need to use an oil based enamel, and you probably won't or shouldn't try to spray it. Instead, buy a can of "Penetrol" at the paint store and use it according to the label instructions. Penetrol makes the paint flow better to hide brush marks. Use a good ($$) china bristle brush, and paint from dry to wet with gentle light strokes. Applying two thin coats is better than one thick one, and watch for drips.
As for the outside, I don't believe in exterior wood doors, because they are high maintenance, and any change in a door's shape will change the way it works, almost always for the worse. Don't just use any old paint, and do whatever you can to protect the door from actual exposure. (Build an awning or something over the door to protect it.) As far as paint, sherwin williams makes a good exterior brand called "superpaint" which is good. I might recommend an oil based for that as well, but ask the experts at the paint store. Whatever you do, do buy the paint at a PAINT STORE and not a "big box" store where the guy who mixes your paint was asking people if they "want fries with that?" last week.