I'd start with the control of the crane, then the control of the claw. That'll be hard enough - especially making it robust enough that it works one time, every time. I hope you understand what usually happens when the machine takes a coin and then doesn't operate? You'll want to make pretty darn sure that this doesn't happen, and if it happens, that the machine can handle the consequences.
When that works you can start adding the extra buttons, the coin op, sounds, etc.
Do those things really use servos? Not regular motors with end stops?