Take everything off the walls nearby to make sure you don't break anything during the demolition.
The posts and railings (image 1) need to come off first so remove the wood plugs hiding the screws. Find and remove the screws under the railing going into the post. The first newel post comes off, and the balusters up to the next post come out easily. Remove the balusters and railing to the next post. Unscrew posts from the railing (image 2). Sometimes screw heads can break off. If that is the case, carefully use a reciprocating saw to cut through pieces to take them out. Typically railings that end in a wall are secured with a lag bolt (image 3). Remove the lag bolt and the rest of the railing.
Slip the pry bar under the top tread and pull it up. Note that removing treads and risers is hard work because not only are they nailed down, but they're glued down as well. Some of the risers will come off with the treads. Cutting through the middle of the treads can help when prying them out. Keep the damage to the drywall along the stairs to a minimum.
Sand and paint the new risers, balusters and skirt boards. Painting many of these pieces before they are installed is much easier then waiting until they are installed.
When the stained parts have dried, apply a coat of polyurethane on them. Use a quality brush, and always follow the grain. Be sure to stir the polyurethane with a stick. Never shake the can because it creates bubbles that will be brushed on to the wood ruining the finish.
To get proper measurements for the skirt boards, you'll need to clean up the stringers first. Cut away the overhang of the landing with a circular saw and hand saw, and remove any leftover frame pieces with the reciprocating saw. Trim up the drywall edges and remove any dried glue. If the new treads are a different thickness than the ones removed adjust for the difference on the stringer. Measure the distance from the floor to the landing and divide by the number of steps. This results in how tall the risers need to be. For this particular project, the stringer treads need to be shimmed 1-1/4".
To mark for the skirt boards, hold a level across the new tread shims and use a framing square to mark the width of the skirt board. Make sure the square is flush with the level on one leg so it is measuring at a 90-degree angle from the line along the stairs. Draw a line where the skirt board will go. And mark for the one under the landing. On the wall side of the stairs, mark a line on the wall parallel with the steps. Then using a speed square, calculate the height of the skirt board.
With the outside skirt board temporarily held in place with a couple of screws, mark the rise and run on the inside face of the skirt board. Mark the inside skirt board with the measurements from step 9. Take the boards outside and cut out as much as possible with a circular saw, then finish the cuts in the corners using a jig saw. The outside skirt board cuts for the risers are made on a 45-degree angle, which allows the risers to have a professional look. Make all the 45 degree cuts first then go back and cut the 90 degree cuts for the treads. All cuts for the inside skirt board are at a 90-degree angle to the surface of the skirt board.
Set the inside skirt board in place, making sure it goes behind the carpet at the floor. Attach it in place with 2" finishing nails into the studs in the wall. Line up the outside skirt board and nail it to the stringer. Use a temporary riser with a 45-degree angle cut on the end as a guide. Use it to make sure the finished risers will line up. Add the skirt board under the landing.
Measure the risers from the wall to the outside edge of the skirt board, and then cut them to length, mitering the end at a 45-degree angle. This will match up to the angle cut on the skirt board. Add wood glue to the joint and make sure it's flush. Then nail the 45-degree joint with a brad nailer. Swap over to a nail gun and secure the face of the riser to the stringers with eight penny finish nails. Do each riser like before working down the stringers. When they're all on, fill in the nail holes with vinyl spackle.