Sand the veneer with 100-grit sandpaper on a random orbit sander. This pass smooths any visible roughness in the grain. Do not oversand with the 100-grit sandpaper. Smooth the edges and work in the direction of the grain. Make a second pass with 150-grit sandpaper, working in the direction of the grain, to align the grain so it will accept the stain evenly. Rub the surface with a tack cloth to remove the dust.
Apply two coats of polyurethane. Use a medium width soft-bristled brush and work in the direction of the grain. Spread the finish as smoothly as possible, working in slow even strokes to prevent drips and runs. Work slowly and avoid quick stops and starts, which cause bubbles in the finish. Allow the first coat to dry for at least three hours before applying the second coat. If you used stain and poly all-in-one for the first coat, use clear polyurethane for any subsequent coats. Apply the second coat in the same manner as the first and allow it to dry for three hours.
Lightly sand the surface by hand with an emery cloth to smooth the finish. Rub the dust off with the tack cloth. Apply a third coat of polyurethane to give the veneer a deeper shine.