Painting a unfinished solid wood exterior door
To do this right take the door down.remove hardware,  remove gaskets. lay door flat on saw horses, apply a  exterior oil based primer to all raw wood with a natural bristle brush 2.5-3”.Allow to dry
Rehang door and wait 24 hours. 

Remove door again lay flat on saw horses. Sand the primed wood carefully smoothing the raised grain. 

Dust off or blow off all  surfaces. Paint with high quality exterior 100% acrylic paint of the desired color and sheen. Use a real nice nylon brush like a Purdy 3” $20.00-$25.00

Let dry. Rehang door. Wait 24-48 hours. 
Remove door. Repeat the sanding and blowing off dust. Apply second coat. Let dry. Rehang door reattach all hardware after 24-48 hours. 

During each painting step be sure to coat the top and bottom of door.

After priming frame and jambs apply wood putty to any holes allow it to dry then sand. Apply caulk to any gaps after primer has dried. Allow caulk to dry before painting. 

After final coat has dried for 72 hours reinstall gaskets.