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As a home builder for 13 years, my opinion is trim first. It makes a much cleaner line where the trim meets the wall. It's much easier to cut in as well. Caulk the joints and where the trim meets the wall. As long as you are using latex on the walls, any specs will wipe off with a damp cloth. Don't be too proud to tape either. For an amateur, you will get a much straighter line and it will actually take less time than cutting in by hand. Go to any paint store and get a taping device. It also catches all of the specs from the roller so it won't get on the trim.

New construction - tape off and mask the house (floors, cabinets etc.) , primer all, caulk , paint ceiling, spray trim, cut in and roll walls
Existing home - tape and mask floors, paint trim, cut in and paint walls
Kirk