Piloting
A crew member who is controlling a helm gains one point of experience for each incoming projectile that is dodged during combat. This includes asteroids, so long as you are in combat at the time. (Note that the dodges do not count while you ship is under the effects of +60% evasion from a cloak.)
This can be explicitly trained by finding an enemy that cannot break through your shields, and letting him fire at you. Increasing your dodge chance (i.e. more power to engines) will help this occur quicker, but even with the default chance you can max out the skill eventually.
Engines
Exactly as for piloting, a crew member manning the engines gains one point of XP for each projectile evaded while uncloaked in combat - and so it can be trained in the same way.
Weapons
A crew member who is manning the weapons station gains one point of experience for each weapon that is fired. It doesn't matter whether it hits or misses, or whether it can do damage (e.g. a beam weapon fired at a shield).
(Multi-shot weapons such as burst lasers still count as a single fire order and hence a single XP.)
This can be trained by firing non-lethal weapons (e.g. ones that do ion damage) repeatedly against a target that cannot damage you.
Shields
A crew member manning the shields station gains one point of experience for each shield "bubble" that is brought back up during combat. It doesn't matter why the shields were down, be that direct enemy fire, ion damage, asteroid strikes, or even manually removing and then restoring power to the shield systems.
This can be trained by letting a target that cannot break through all of your shields repeatedly take one layer down. (For fastest results, move your pilot away from the helm, or reduce your engines to 0 power. This will reduce your dodge chance to 0%, ensuring that everything hits your shields.)
Note that you used to be able to manually remove power and restore it in combat, to get points, but this has been patched.
Repair
A crew member gains one point of experience for getting the "finishing blow" when repairing a (sub)system. Note that hull breaches provide no repair experience.
There's no specific way to train this as it requires your systems to be damaged. Though you can focus the XP gains by always having the same person run around and repair everything after a battle.
Combat
A crew member gains one point of experience for getting the finishing blow when fighting an enemy crew member or (sub)system.
This generally gets trained in "real" situations. However, since XP is gained for taking out systems, a few extra XP can be gained from boarding if one tries to take out the enemy's systems before the crew.
Killing the crew in first phase means they won't be there in the next two, same if you kill them in second phase: they won't be there in the final, so killing them can be a good tactic as they may trouble you later.
If you kill them all you won't win that phase or overall, instead the ship will go into auto-pilot and will slowly repair all its systems much like the auto-assault scout drones in earlier sectors. It's best to leave one enemy crew alive, maybe the isolated guy on their laser tri-shot; because fighting an AI Boss is much harder than fighting a single panicky engineer.
When the ship flees to its next phase your boarding crew will be killed if they are still on board, same if you destroy it completely with your crew on board, meaning their noble sacrifice won't go down in the credits.
Best Answer
From this page:
Humans comprise the majority of both the Galactic Federation and the Rebel Fleet. Humans learn skills faster due to their reduced XP ability, have no notable weaknesses and are the cheapest race to hire from Stores.
Engi originate from the Engi Homeworld. They take 50 to hire, with the following special traits:
The Mantis originate from the Mantis Homeworld. It takes 45 to hire them.
Rockmen are strong creatures and take 65 to hire. Their abilites are:
Zoltan usually appear in Engi Homeworlds or Engi-Controlled Worlds. They appear rarely, and it takes 65 to hire them. Their special abilities are:
Slugs are one of the rarest crew to find. Their advantage is the ability to see your enemy's crew, and to be able to see the rooms surrounding the slug in your ship if your sensor goes down. They cost 45 to hire.
The Lanius are scavengers – metallic lifeforms that rely on the absorption of minerals to sustain themselves. They are available only in advanced edition and even then are almost exclusively found in the Abandoned Sector where they cost 50 to hire. They need no oxygen and also drain oxygen from the room they are in.
And a final 'secret' race: