Before post-processing any of the parts, I did a dry fit to see if everything lined up. No sense in finishing a part when it doesn't fit. I normally would 3D print the part first to do test fitting, but the tolerances are much different than CNC milled parts. PLA and ABS tend to expand when it's being extruded, so the layer lines contribute to different variable thicknesses. I found a 0.1mm offset vertically to be a good value for getting a friction fit.
There was a lot of burrs along the edges of the wooden parts. To remove them, I sanding the edges down using 320 grit sand paper. For some of the cutouts, I used a filing tool. A sanding/griding bit for a dremel tool is also a good option.
I didn't apply any solutions to the surface, but it's definitely an option worth exploring.