How much force is produced by control surfaces?
From the OP, it seems like you are treating the aircraft as a point mass. As a point mass, you don't get the rotational degrees of freedom, so control surface can't be accurately modeled. In the case of takeoff, rotation is achieved via elevator up force, resulting in aircraft pitching up, increasing its angle of attack, resulting in increased the overall lift and achieving the initial vertical acceleration. In summary, you can't model it from first principle without the rotational degrees of freedom.

There are a few things you can do. 



You can continue to assume point mass. You can assume some simplified parametric model for takeoff rotation. That is, the trajectory is a parametric relationship of airspeed, time and elevator. It's not a physical model and it won't be accurate to the magnitude of elevator application, but it may suffice. Once in the air, you can do the same thing with banked turn for aileron applications, and airspeed changes with elevator. Essentially, you are working with a performance model (see performance equations of motions), with artificial relationships for control surfaces.

The above may suffice if all you want to do is GTA style flying (as an aside, its takeoff behavior is terrible). It'll look like a plane flying, but won't feel like a plane flying to anyone who's flown a plane.



You can move up to six degrees of freedom. In that case, you'd need more than just lift, drag and thrust; you'll need moment relationships as well. I recommend you to give Etkins, Dynamics of Flight a read. You can skip directly to Chapter 4. You'd be in very good hands once you've read Chapter 4 and 5.