John Willis and m Ransley,
Where did you learn the technique to use multiple coats of primer?
Primer's function is as a sealer and bonding coat. It is purposefully
thinner to soak in rather than sit on the surface. It seems to me that
once the surface is sealed there isn't much reason to seal it again.
At that point you want to start filling voids and adding durability and
color - in other words, the top coats.

To the OP:
I just noticed that you didn't specify what finish your paint is. I
trust it is semi-gloss or gloss. Those finishes will give you a
smoother finish than something more matte. I also neglected to mention
that removing the door from the hinges and laying it on some sawhorses
so you can paint it flat will allow the paint to lay down better so
there are no brush marks. Just watch out for the drips over the edges
and clean them up before they set up.

Contrary to popular opinion, sanding acrylic paint works. Latex paint
is softer and doesn't sand well. Sometimes the two get confused. You
will have to make sure that the paint is totally dry before attempting
to sand. You're not relying on the sanding to smooth out the surface
so much as knock down the nibs and high spots a bit in preparation for
the next coat. The final coat does not get sanded, so you need a dust
and draft free space to work in.

R