In addition to my small basement shop I'm a member in the local makerspace. The Columbus Idea Foundry has a ShopBot CNC Router and a 60W Laser cutter in addition to a complete wood shop with the usual compliment of stationary, hand and power tools. Given all the small circles and material being used I decided to fab the parts on the laser. The Mini 24 has a max size for the 12" x 24" which was more than enough for the largest piece. The only other tools used were a flush cut pull saw and a sanding station.
It might seem a bit backwards material selection was first on my list. I think in a project like this since it helps to prevent surprises down the road. The material needed to be tough, durable and easy to work with and take a kid friendly finish. A quick trip to Woodwerks and I've got a sheet each of 1/4" and 3/8" Baltic Birch plywood, 1/8" and 1/4" hardwood dowels and a fresh bottle of glue.
The imperial dimensions of the plywood are approximate, actual dimensions are metric so 1/4" is nominal 6mm and the 3/8" is nominal 9mm. Actual measured thicknesses are: 5.96mm & 8.94 respectively.
First step on the laser is to determine accurate settings for the finger and mortise joints. The kerf of the laser is small but needs to be accounted for in the design to have proper clearance for a good glue joint. A couple quick test pieces finds 0.23" is correct for the 6mm ply and 0.35" for the 9mm.