Well, with the primer it all depends on what you're going to use for paint. You absolutely can NOT use a latex based primer and then use an oil-based paint on top. It's just not compatible and you'll have nasty results if you do.
Many years ago, oil based was always the no-brainer if you wanted a durable, smooth finish paint. Thankfully, latex technology has gotten so much better these days that you can paint a cabinet with latex and have it be just as durable as with oil based.
Latex paints tend to dry quicker and have far less nasty fumes than oil based paints. Clean up is also a big plus since latex paints can typically be cleaned with water.
The biggest thing to remember regardless of which type you use is the environment. You do NOT want to paint in a dusty area, nor do you want to paint in an area that has poor ventilation. Good ventilation allows even drying/curing of the paint and dust free ensures that no dust particles settle on your wet paint job.
Before you paint, it's a good idea to thoroughly sand your cabinet with a mid-grade sand paper to create a surface that the paint can adhere to. When done with that sanding, use a tack cloth to make sure you have removed any dust left on there.
Whether it's primer or the final paint, multiple thin coats are far better than one thick and sloppy single coat. I personally would put on two or three coats of primer, sanding in between each coat to ensure a smooth finish. If after two coats of primer you have sanded it smooth and not sanded through to the surface below, then you can apply thin coats of paint, sanding in between, until you get the color saturation you're looking for.
Always sand lightly in between each coat of paint. This helps ensure a smoother overall finish.
A final clear coat can help seal everything in and allow you to buff it out and get a great gloss finish. However, read the paint can very carefully. Make sure you get a paint that WON'T yellow over time as this will make any paint job you've done underneath look like crap. And, if you are putting artwork on the cabinet, MAKE SURE THE PAINT IS COMPATIBLE!!!!!!!! Water based clearcoats are typically more compatible than the other solvent based clears, but always double check this. Nothing worse then spending hundreds of dollars on great artwork and weeks on painting and sanding only to ruin it in five seconds by putting on an incompatible clear coat.
One more thing is to allow ample drying time between coats. If you paint one coat on the weekend and don't get back to it until the next weekend, that's great. This allows any solvents in the paint to gas out and cure so that you don't get odd bubbling later on down the line. This is especially important after you adhere the artwork to the cabinet. Let it cure and degas for a few weeks just to play it safe before you clear coat it.
The biggest advice is to take it slow. Don't rush it. This will be the most noticeable part of your cab so it's important to take it slow and do it right.