Finishing a shed with paint and caulk is important if you want your shed to last longer than normal.
Before painting your shed, I recommend that you caulk with a paintable silicon caulk. Yes, that's right, they do sell paintable 'silicone caulk' now! Caulk along all exposed horizontal joints where water can get in, and also caulk along vertical joints where water can work its way in.
While you are building your doors, it will be easier to caulk as you install each peice of your door rather than later. It takes a little more time to do it this way, but it will insure that your doors are properly caulked.
Any bare wood needs to be painted with an exterior wood primer first before painting with a finish coat of paint. As far as painting with oil versus latex paint, it really makes no difference. If you paint with oil paint and later on decide to recoat with a latex paint when its time to recoat your shed, just remember you can't paint over oil paint with latex unless you apply a primer coat first over the oil, then recoat with latex.
Although it looks nice, I don't recommend putting any horizontal wood trim along the bottoms of your siding. This is just asking for trouble over the years as water will eventually accumulate under these boards and cause the bottom of your siding to rot out.