My favorite primer for wood doors is Bins Bullseye, pigmented shellac. This primer bonds well, hides wood grain well and gives a very smooth surface for top coats. If you prefer to stay with a latex product, then Kilz premium is good too.
Remember prep is of the utmost importance for a good finish. Sand everything and clean it well before applying primer. On a new piece, I like to lightly sand the first coat of primer. You will feel the difference, trust me. I like to use two coats of primer, then sand it again very lightly to make it super smooth: 220 grit used lightly is fine. Clean it well with a tact cloth or very slightly water dampened rag.
Now for the top coat. Use a premium quality 100% acrylic exterior trim paint. Usually a gloss or semi gloss looks great, but that's a personal preference. Don't skimp on the brush: get a good Purdy Glide, poly/nylon 2½" angled, soft chisel tip. Apply nice smooth, thin coats, and expect to do three coats for a good solid fill. Start in the panels and work your way out to the rails and stiles. If you can take the time to remove the door after it is installed, (never remove door from frame before installation!!!!) lay it flat to paint. Laying it flat will really help avoid drips and runs. Pay special attention to the panel corners and details for drips and puddles a few minutes after you apply the paint. Use long smooth strokes to finish each panel, rail and stile.
BTW, a really good latex 100% acrylic is every bit as durable as an oil based product. Oil based paints are going away and have been compromised greatly in recent years due to VOC laws. I stopped using oil paint several years ago. The only oil based product I use now is urethane.
On numerous occasions I have had to cut hollow core doors off more than the available blocking in the top / bottom of the door. What I have done in these instances is to take the cut off bottom piece and then reclaimed the inner block for re-use. If a table saw is available it is quick work to set the fence just right so that a couple of rip direction passes cleanly cuts off the outer laminations. Alternatively the laminations can be removed using a plane.
Once the block is nicely cleaned up it can be glued and slipped right up into the open bottom of the door. Use a couple of 1x4 boards on either side of the door and some good sized C-Clamps to tighten up the glue joint so that the glue can dry without any gaps.
Note that often the inside of hollow core doors has an internal webbing of cardboard material to bridge the gap between the two lamination skins of the door. Sometimes it is necessary to use a chisel inside the cut open bottom of the door to scrape back the glue joint of this web material on the laminated panel. If this is not smoothed out the re-claimed bottom block will make the laminated sides bulge out when trying to glue the block in place.
Best Answer
You won't get cheaper or easier than two layers of sheet lumber. Particle board is a common component in 30-minute fire doors (though they usually have a hardwood veneer).
I'd do just as you plan. Sand or route a bullnose all the way around and seal it with a couple coats of urethane, smoothing with steel wool between.