How do wings generate lift?
To get to the bottom of it, it might help to look at lift at a molecular level:
Every air molecule is in a dynamic equilibrium between inertial, pressure and viscous effects:

Inertial means that the mass of the particle wants to travel on as before and needs force to be convinced otherwise.
Pressure means that air particles oscillate all the time and bounce into other air particles. The more bouncing, the more force they exert on their surroundings.
Viscosity means that air molecules, because of this oscillation, tend to assume the speed and direction of their neighbors.

Flow over the upper side of the wing
Now to the airflow: When a wing approaches at subsonic speed, the low pressure area over its upper surface will suck in air ahead of it. See it this way: Above and downstream of a packet of air we have less bouncing of molecules (= less pressure), and now the undiminished bouncing of the air below and upstream of that packet will push its air molecules upwards and towards that wing. The packet of air will rise and accelerate towards the wing and be sucked into that low pressure area. Due to the acceleration, the packet will be stretched lengthwise and its pressure drops in sync with it picking up speed. Spreading happens in flow direction - the packet is distorted and stretched lengthwise, but contracts in the direction orthogonally to the flow. This contraction is needed to make space for that wing; in supersonic flow it will decelerate for the same purpose. Once there, it will &quot;see&quot; that the wing below it curves away from its path of travel, and if that path would remain unchanged, a vacuum between the wing and our packet of air would form. Reluctantly, the packet will change course and follow the wing's contour. This requires even lower pressure, to make the molecules change their direction. This fast-flowing, low-pressure air will in turn suck in new air ahead and below of it, will go on to decelerate and regain its old pressure over the rear half of the wing, and will flow off with its new flow direction.
Note that lift can only happen if the upper contour of the wing will slope downwards and away from the initial path of the air flowing around the wing's leading edge. This could either be camber or angle of attack - both will have the same effect. Since camber allows for a gradual change of the contour, it is more efficient than angle of attack.
Flow over the lower side of the wing
A packet of air which ends up below the wing will experience less uplift and acceleration, and in the convex part of highly cambered airfoils it will experience a compression. It also has to change its flow path, because the cambered and/or inclined wing will push the air below it downwards, creating more pressure and more bouncing from above for our packet below the wing. When both packets arrive at the trailing edge, they will have picked up some downward speed.

Behind the wing, both packets will continue along their downward path for a while due to inertia and push other air below them down and sideways. Above them, this air, having been pushed sideways before, will now fill the space above our two packets. Macroscopically, this looks like two big vortices. But the air in these vortices cannot act on the wing anymore, so it will not affect drag or lift. See here for more on that effect, including pretty pictures.
Lift can be explained in several, equivalent ways
Following the picture of a pressure field outlined above, lift is the difference of pressure between upper and lower surface of the wing. The molecules will bounce against the wing skin more at the lower side than at the upper side, and the difference is lift.
Or you look at the macroscopic picture: A certain mass of air has been accelerated downwards by the wing, and this required a force to act on that air. This force is what keeps the aircraft up in the air: Lift.
If you look at the wing as a black box and only pay attention to the impulse of the inflowing and outflowing air, the wing will change the impulse by adding a downward component. The reaction force of this impulse change is lift.
Either way, you will arrive at the same result. By the way: Most of the directional change happens in the forward part of the airfoil, not at the trailing edge!

Lift is a matter of definition
Lift and induced drag are both part of the pressures acting on the wing. If you add up all the pressure forces acting on a wing, their resulting vector will point slightly backwards. The streamwise component is drag, and the component orthogonal to the direction of movement is lift. This is just a defininion, made for simplicity.