Is the Angle of attack of a plane simply the angle of the wings?
No, the airplane rarely flies "straight forward". It's usually pitched slightly up or down. So the angle of relative wind to chord line of the wing varies. The chord itself can also have an angle relative to the airplane's body.
If you are actually interested in writing a flight simulation, you need to understand how the forces acting on the airplane produce an equilibrium pitch, and therefore equilibrium AoA for the airplane, and how the pilot can control this, because this is how the airspeed for the aircraft is actually adjusted. (Throttle is used in uniform flight to adjust rate of descent/ascent.)
At a minimum, you need to consider 4 aerodynamic surfaces. Left wing, right wing, horizontal stabilizer, and vertical stabilizer. That last one is fairly straight forward, but the first 3 play very important role in balancing the aircraft. The center of pressure of the wings is located behind center of mass and generate drag (against relative wind) and positive lift (perpendicular to relative wind). Because the force vector points up, the torque from wing lift tries to pitch the plane forward. To prevent that from happening, the horizontal stabilizer provides negative lift. It pulls the tail down. It also generates drag, same as any other surface.
If an airplane is traveling at target air speed, the torques from wings and horizontal stabilizer balance each other out, and the airplane continues at whatever angle of attack it currently has. If the airplane travels faster, the lift exceeds weight, airplane gets some positive vertical velocity, which means relative wind gets some negative vertical velocity. The angle of attack reduces, which means less positive lift on wings and more negative lift on tail. The aircraft pitches up and slows down. If the aircraft is traveling too slow, it begins to descend, pitches down, and speeds up.
So how do you adjust the target speed? That's where control surfaces come in. You don't just have a fixed wing. You also have ailerons, elevators, rudder, flaps, slats, and numerous tabs. If you do not need extreme precision, there is a very simple way to simulate control surfaces. You can simulate them as the entire surface rotating slightly, changing angle of attack. For small deflections, it's actually fairly accurate, so it will work for ailerons, elevators, and rudder at least. Also, for the basic simulation, you can forget about the fact that you have other control surfaces, and pretend that ailerons run the length of the wing. This way, you still have just the 4 surfaces mentioned earlier, each of which you can control the AoA slightly.
So what do you do? Well, if you want to reduce target speed, you need to make equilibrium position slightly pitched up. For that, you reduce AoA of horizontal stabilizer, increasing its negative lift. This corresponds to pulling on the yoke/stick slightly or adjusting the trim tab accordingly. And vice versa. If you want to speed up, you increase AoA of horizontal stabilizer reducing its negative lift.
The rest is straight forward. You adjust AoAs of wings to get rolls, and you can adjust AoA of rudder to make small corrections or to get coordinated turns. If you program drag correctly, that is, make sure it's related to AoA and lift, you should get all of the associated effects gratis. All in all, such simple consideration gives you a fairly descent flight sim. (I've written a few implementations of this, it works.)
