Barton's advice all good- he's been there and done that.
Eddie has been there and built that
We took simple 1x3 and 1x4 and some thinner pieces of pine, did a quarter inch route on the corners, and use that for trim. It matched the theme of the house (simple country, square corners). This was also driven by function- I didn't want to be tied into some custom mill work that I might not be able to get in the future.
We primed all the wood before installation to save time on painting after the wood was cut and installed. It was pretty easy to set up the trim and prime it ourselves after the routing had been done.
On caulking, after seeing the painters caulk everything in part of the upstairs, they were removed from caulking duty. The intersection of the drywall and trim had a roundness to it (the radius of a finger) and taping it off to paint the trim was impossible. I ended up only caulking the top of the base boards and it was flush/90 degrees with the wall. Note: trim color was barely off white and matches most white caulks. I caulked this because the gap is something you really see because of the viewing angle. Didn't caulk anywhere else because eventually the caulk will crack and that's one more thing to maintain. The house has foam insulation so I did not depend on the trim for sealing the house. At the risk of sounding like a bad contractor, once you're moved in and living there you don't notice the no caulking on the trim.
Our sequence of trim/painting during construction was to clean and mask everything, spray the primer, roll the color on the walls, then install trim and paint it. This made for a clean color transition between the trim and the wall. We also ended up painting the trim because the painters did not fill the nail holes as per Barton's comments. The also had the trim paint too thick, so there were brush marks. I thinned the trim paint and did two coats. On window sills, I did the 2nd coat were the trim paint had a lot of thinner it it for a very smooth surface.
Only down side to all this is white trim shows dirt more, and the flat top on the baseboards means you have to wipe or vacuum the trim every 3 months or so (and the cats don't help here either).
So for me the advantage of not caulking was less time, don't have to worry about movements and cracks, clean sharp boundaries between trim and wall both in texture and color.
Disadvantage is you will see some small crack or gap. How the job is being done (order of operations) is important. If it the cracks really bothered you, you could go back and caulk and touch up. And this is not necessarily a unilateral decision. If a few gaps on some doors or windows are just too much, or at an angle where you really notice the small gap, you can just caulk those. I had 3 places where this happened.
Sorry that the vector sum of a lot of the advice here is often zero, but I think that is due to the variety of construction practices out there.
Pete