I finally got a chance to put a few coats of primer on the cabinet. There's nothing special about my technique - I am using Zinsser BIN Primer (pictured) and so far I've put on 4 coats using a foam roller and a paintbrush (for the inside corners). I sanded with 400 grit sandpaper in between coats and the entire surface is fairly smooth. It isn't perfect by any stretch and there is some "orange peel" effect over the entire surface. I still have to do some final sanding before putting the top coats on.
Here is where I'm at:
I should probably get a shot of it outside instead of in a dark garage but whatever. I didn't take many pictures of this process since it is hard enough to do without worrying about my camera. Plus, this part is boring.