As a far as I can tell, Mechwarriors earn xp by how many kills they get per mission. It might also include shots that connect with each target. Depending on the number of Gunnery and Piloting skills combined they rank up from Green all the way to Ace. So if your Mechwarrior has a combined number of 110 they will be promoted to Regular. If a Mechwarrior is airborne(jump jets) and makes a kill or connects with a target, they gain more piloting xp. If a Mechwarrior makes a kill or connects with a target on the ground, they gain more gunnery xp. In my experience that's how I was able to get Ace rank Mechwarriors.