Remove the existing railing. The railing should be attached to the wall by brackets. With an electric screwdriver, remove the screws holding the brackets in place. Start with the brackets on the edges and move inward so that the weight of the railing remains evenly distributed. Tap a finishing nail into the old screw holes to make sure the original railing was attached to the studs. If it was, this same foundation can be used for the new railing. Never attach a railing to plaster or drywall; it won't be secured.
Drill new screw holes to ensure a sturdy installation. Mark three lines 1/2 inch up from the bottom screw holes and punch a small hole in the wall at each 1/2-inch mark using a nail set. The brackets for the new handrail will be installed where this punchmark is. Cover the old screw holes with spackle.
Measure the stairwell to determine the length of the new handrail. Use a circular saw to trim the edges at a 45-degree angle. Cut it so the pointed end of the 45-degree angle is at the top for the top part of the railing and at the bottom for the bottom of the railing. With your wood glue, attach the trimmed pieces to the opposite ends they were cut from, so that it forms a 90-degree angle that will lay against the wall. Because oak is a hard wood, drill pilot holes before securing the glued pieces with two finishing nails.
Attach the brackets to the wall using wood screws and a screwdriver. The bottom screw should be placed where the punchmark was placed. Once the brackets are in place, set the railing on top. Drill pilot holes before attaching it to the wall brackets. Finally, use wood screws to attach the railing to the brackets.