Types of different plywood core
Your veneer should be sanded to 180 prior to applying the stain. Use a washcoat, Mix one part sanding sealer to one part lacquer thinner and spray a wet coat, let dry, then sand with a 220-grit stearated paper or 3M fine-grit sanding sponge prior to staining. You will be amazed at how even the color looks after a washcoat and proper whitewood sanding.
I also recommend using an N.G.R. dye stain and/or toner prior to the washcoat to add depth to the finish and even more color uniformity.
Our supplier has prefinished panels as well that are factory coated, you may want to call around to see if you have that choice in your area?