I would lightly sand the existing enamel then use a tack cloth to remove dust. No need to prime. Use a good brush.

As to your doors, I much prefer working on them flat. Same prep as mentioned above. Remove hardware. Paint with the grain. Paint a section at a time. For long pieces, use long brush strokes. Don't apply too heavy but do two coats. After one side is dry flip and repeat. Good luck.

Should I use a brush for the whole thing or can I use small rollers with a 3/8" nap? Thank you for your response!

I'd be fearful of texture with a 3/8 roller. What you want to prevent is paint runs. I personally have more control with a brush when doing trim. Try a roller but have a brush handy just in case.

I tried this on the inside of my hall closet door. I painted the insets of the panels, then the flat middles of the panels, then the rest of the door, all with a brush. I let it dry and did a second coat the next day with a brush, but I used a foam roller to eliminate the brush marks. It's almost completely dry right now and it looks amazing! This is a lot of work though, 7 more doors plus a pocket door and 2 sets of bi-fold doors.. this may take a while. I wish I had a paint sprayer.