Sounds like your score motor might not be traveling home correctly or is detecting home position too early (and forcing on the start relay returns it home, which is why it does that the first time then works the second time), adjust/clean the home position switch on the motor, there should be a score motor switch chart stapled on the inside of your cabinet and if that's missing it's on the schematic.
My Domino does the same thing after I had to rebuild part of the score motor, going to have to go in and redo it.
Otherwise there are a few things in the path of the start button to the start relay, such as the switch on the credit unit. Your game isn't set for free play by closing the switch on the credit unit yet? Do you prefer coins?