What's the best way to have the flat cylinder plate follow along the path(s) drawn, and then have those two vertical cylinders above simultaneously orbit around the empty?
Ok, here's what I have so far. I'm still trying to work on the keyframed rotation start, but the object keeps staying in the same place when everything else keeps moving.
Here is the base setup. I used a simple flattened cylinder as a disc, two bezier circles, one of them scaled by 10 in edit mode, and another cylinder I made skinny and transformed/duplicated in edit mode so it becomes one object comprised of 2 cylinders sitting on either side of the disc. I made them all in the center because that way I could be assured they all have the same origin.
I parented the small bezier circle to the flattened disc, and applied a Path Follow Constraint to the two-cylinder object, pointed at the small circle. I set it to follow curve, and noticed that this changes the transform for some reason. I simply went to edit mode, selected everything, and moved it back over the flattened disc (for me it was simply a 1 unit transform on the Y, I think). Anyway, once it was back over, I Animated the path.
Then I selected the flattened disc and added a Path Follow pointed at the larger curve (which represents your main path). I checked Follow Curve, Animated the path, and presto - two cylinders rotating around the center of a disc that is moving around the scene.
I know this isn't the final result you're looking for, but you requested my steps up to this point, so there they are.
I'll upload the blend, in case you want to have a look. It's nothing special. I'll also keep working on how to keyframe the cylinder rotation start delay (assuming that's the part you want delayed). I'll post an edit to this answer if/when I find it.