There's not "facing"... just think of the flying movement exactly as you think of ground movement, except that you have a slight restriction in how fast you can corner.
On the ground, you move in a direction. You get to turn as you see fit, but you still move in a direction as you count out your squares. In the air, in this example, as you count out each square you can go only into 3 squares: the one directly in the same direction as you were traveling, or one of the two next to it.
It's really as simple as that.
There's not "facing"... just think of the flying movement exactly as you think of ground movement, except that you have a slight restriction in how fast you can corner.
On the ground, you move in a direction. You get to turn as you see fit, but you still move in a direction as you count out your squares. In the air, in this example, as you count out each square you can go only into 3 squares: the one directly in the same direction as you were traveling, or one of the two next to it.
It's really as simple as that.
Hmm... My brain is unfortunately still not grasping this.
What I don't really understand is how a creature turns while flying without taking a test.
You don't, unless you have a feat, class feature, racial ability and/or something else that allows you, under certain conditions, to manuver without making fly checks. This info will be described in the feat, class feature, etc.
How do you perform a U-turn in mid-air?
Best way is with the Wingover feat, but you want to do it using a creature with a 45 degree turn radius. Here's the deal, while there is no longer a concept of 'facing', there is still the concept of direction when you are moving. Flying a certain distance allows you to turn a certain number of degrees based on your manuverability.
EX. Initially flying North you want to execute a 180 degree turn with a 45 turn radius over 5 feet for Avg manuverability. To execute the turn you move 5 ft forward, but to the square 45 degrees off of your center (we'll go west). you are now pointing (facing if you will) to the NW. For your next 5 ft. you move one square 45 degrees off of NW, which is now has you facing (pointing) due West. Your next 5 ft. you move 45 deg off of West to point South West. from SW you move another 5 ft @45 deg. off of SW to South. Voila, you have just executed a 180 degree turn, consuming 20 feet of movement. And yes, the whole while the space you occupy does 'snap to grid'. This isn't a problem because there are no facing rules.
If you can only go in the three squares ahead of the direction you came then you can't seem to turn at all, or at least without making a lot of tests or waiting till the next turn when the direction you were heading in is reset.
Yeah, you missed the point, your 'direction' doesn't reset; without facing rules you can face any direction you like, and you can use direction to navigate and execute your movement and turning while in flight.
EDIT: I probably screwed up some of the rules up there, particularly the amount of movement consumed during the execution of turn, but the example still gets the point across regarding the basics of how to turn.
Actually, the PRD is pretty succinct on how to fly
http://paizo.com/pathfinderRPG/prd/skills/fly.html#fly.