The need for primer depends if the doors and trim are factory primed or not. If a primer is already applied to these items there isn't a need to re-prime. Raw wood will require primer but the doors will not.
For unprimed wood trim I like to use an oil based undercoater. This type of primer dries quickly and is easy to sand, provides a very slick surface for a smooth finish. An alternative is Kilz Original Oil Base primer, you will need to apply a little more Kilz for the same affect but it works good.
Undercoaters are available in oil base and acrylic formulations. Both work equally well, I just prefer the oil base for its faster curing.
The type of paint you use depends on your needs. Oil based paints cure harder and are less prone to damage, but smell bad when painting and take forever to cure (typically 2-3 weeks). Acrylic paints are much easier to use, cheaper and dry much faster. Plus a very good acrylic paint is almost as tough as oil base.
Typical paint sheen's applied to doors and trim are satin or semi-gloss. As a minimum, a satin sheen should be used. The sheen of your trim and doors is a matter of personal preference, but the shinier to finish the more flexible the paint is and easier to wash.