As a physics newbie I'm trying to get a grip on basic orbital mechanics. I think I'm beginning to get grasp on how bodies interact with each other. When a body approaches another body it accelerates due to gravity. It can reach a point where its velocity is high enough to keep falling but also keep missing the object it is falling towards. What keeps it from accelerating (because of gravity) and eventually reaching escape velocity? I feel like I'm either looking at things the wrong way or I have the entire thing wrong.
It does keep accelerating. Its velocity in the direction of the object being orbited keeps increasing.
But this direction keeps changing. The reason the satellite's total speed doesn't increase, at least in the case of a circular orbit, is that while its velocity towards the object increases, its tangential motion moves it forward so that that direction is always perpendicular to the direction of motion. Thus while the satellite is undergoing constant acceleration, that acceleration is always perpendicular to the direction of motion and the speed of the object never changes.
You're right - the objects are always accelerating while orbiting, and this acceleration is due to gravitational influence.
However, force/acceleration don't always cause an increase in speed. For example, a force acting opposite to the velocity will slow an object, and a force acting perpendicular to the velocity will merely change its direction. In an object moving in a low-eccentricity orbit, the gravitational force is always nearly perpendicular to the velocity, so there isn't a large speed change.
If we draw in the acceleration and velocity vectors of an orbiting body, you'll see why it acceleration due to gravity does not cause an orbiting body to fall towards an object.
First, assume the orbit is circular. In any motion of an object, the velocity vector is tangent to the object's path. Because acceleration due to gravity is straight towards the center of the body, the acceleration vector connects the object to the center of the body being orbited. This vector is perpendicular to the tangent vector because the radius of a circle is perpendicular to any tangent to that circle. Thus acceleration is perpendicular to velocity and cannot change the velocity's magnitude.
But, the object is still technically falling towards the center of the attracting body. The reason the object doesn't hit terminal velocity is because perpendicular accelerations don't change magnitude of a moving object.