The staircase is pretty straight forward. Grap some 1" X 4" pine and cut it into pieces for the flat part of the stair and the vertical part. There are actual dowel sets that you can buy that include the fluted dowels, drill bit, and a rubber guide to put on your drill bit so that you don't drill the hole too deep. For everything you need to know about dowels, see
this site. I would use 2-3 dowels at each joint. For the most strength, I would drill into the butt-end of the vertical piece and then into the bottom side of the horizontal part (so that the dowels are running vertical). Make sure you don't put the dowels too close to the edge, otherwise, Buddy might chew into them while he's re-shaping the stairs.
I could try to draw a picture if I'm not making sense.
