I'm still working on ideas for the climbing components, but since this is a Christmas present for my wife and daughter, I need to have something to show in a few days.   I expect that I'll ultimately rebuild them with baltic birch plywood, but in the meantime I'm using some cheaper 1/2" birch plywood from Home Depot that I had left over from another project. 

Each level in the structure will have a tray base, which will contain and conceal the sisal pad (or similar) that will fit inside.  I'm still puzzling-out the other climbing components that will fit inside the trays and lead to the next level.    Two of the trays will hold an acrylic half-sphere, so I started with with those trays.  With two trays, I'll be able to do a test assembly of the whole thing and see how things are looking.

I applied some iron-on edge banding to the exposed flat edges of the plywood.  I'll just put a round-over on the less-visible circular cutouts.

Each tray will mount to the uprights through a pair of holes in each narrow side of the tray.  I threw together a simple drilling jig to center the holes on the sides.

The jig is two-sided, so it can reference the same long tray edge for each pair of holes (just in case the holes aren't perfectly centered, they'll at least be equidistant from a side).

Once I get some finish on the uprights, I'll do a test assembly.