Priority One

Piloting and Engines: Both increase your evasion which is extremely important during the whole game and during the final boss. Your goal is to have achieved anything above 40% up until the Rebel Flagship. 50% is ideal.

Priority Two

Weapons: Give enough hours to this game and you'll realize that if your weapons had striked a second earlier you might have won the game. The reduced time in weapons makes a huge difference so this priority No.2.

Priority Three

Shields: Reduced time for shields is great so if you want to invest with a crewmember. However, when you need a crewmember to fight or fix something, this is the one you use, not the Pilot, the Engine or the Weapons one.

Rest

Everything else depends on your strategy and the various situations. Sensors and Doors can work great if you know when and how to use them.

It dependes.
It depends on the situation, your loadout, your ship, the opponent, are you boarded, are you in a hazard zone, do you need repairs... You get the idea.

There are 6 systems that benefit from manning. Piloting, Shields, Weapons, Engines, Sensors and Doors.
Piloting is the only one you'll want manned in almost every situation, because without a pilot, you can't evade anything unless hte Piloting is upgraded. Also, without your pilot, your FTL won't charge nor is it activatable, if fully charged.
Shields, Weapons and Engines give you bonuses to shield recharge, weapon recharge and evasion respecively. The choise between these is often what you need to figure out fron the situation. Luckly you'll usualy only need to leave one of these unmanned.
Sensors and Doors are the ones you are most likely to leave unmanned outside special situations. The benefits from those are highly situational.

Other than that, manning stations gives you various bonuses and it's up to you to figure out, which ones you need the most.

You can find more on the bonuses on the Systems and Skills wiki-pages.