Why are fuel tanks in the wings filled first, and why are they used last?
It's for wing bending relief (for cantilevered wings). As the generated lift bends the wings upward, the weight of the fuel will counter that.

As the plane loses weight in-flight due to burning fuel, so does the need for wing bending relief (less weight → less lift), that's why the wing tanks are the last to be used.

For context, a Boeing 777-200ER can hold 29 tonnes of fuel in each wing, and each wing measures ~27 metres and is self-supported.



Example:

Let's say in our airliner each wing tank holds 100 units (200 total for both tanks), and the center tank holds 200 units, and you have a flight that needs 200 units of fuel.


Correct scenario: By filling only the wing tanks, the weight of fuel in the wings will counter the bending of the lift.
Incorrect scenario: By filling only the center tank, the wings will bend a great deal (maybe even beyond their design limit). This is not good, and even if it is within the design limits, the repeating [avoidable] stresses will shorten the plane's lifespan.


Example for order of use (schedule):

Our flight needs 300 units, based on the above, we will fill the wing tanks, and fill half the center tank.


Correct order: By emptying the center first, we have extended the duration of the wing bending relief.
Incorrect order: By emptying the wings first, we lose the wing bending relief, while the plane is still heavy.




Related: How is fuel stored and routed on a trijet with one engine at the tail?