We did not have much flexibility with the new stairs, as the location of the kitchen door and side door could not be changed. Drawing up the plans for the stairs was not too difficult, but I checked the measurements over and over to avoid duplicate work (and I did find a couple of errors along the way). I treated the landing like a big step, so the rise and run are consistent between the upper and lower staircase. It would have been nice to decrease the rise a bit and increase the run a bit, but that just was not possible with the location of the side door and first floor joists.
The first step in building the stairs was to frame the wall beside the stairwell.
Once that wall was in place, I installed the landing using 2x10's and 3/4" plywood. This landing is supported on one side by the framed wall and on the other side by a half wall and some openings in the bricks underneath the door.