all of them not working could be a 120V issue on the coin door. The fuse suggestion was good, but afaik there aren't any fuses in the circuit besides the main power fuse.
the ballast looks like a box, the starter is a cylinder.
usually a white wire is on one end of the tubes. That's the voltage wire from the main power fuse. A black wire is on the ballast ... that's the 120V connection. You'd want to see if you have 120VAC when measuring with the meter on those two points.
you don't want to probe the wire between the ballast and the lamp socket. That wire may have much higher voltage on it.
you can swap around the starters / tubes and see if that helps.
starters are pretty cheap, but ballasts not so much. These days, people replace the tube with an LED tube since that can cost less than a ballast. You typically need to remove the ballast and starter and connect the black wire that was on the ballast directly to the tube socket. Easy way is cut out the ballast and connect the two wires together that was on it, then remove the starter.
pull a tube and get the part number off it, then google for an LED replacement.
you have other machines to borrow parts from?