Action-Reaction pair in free body diagram

When you draw a free-body diagram, you actually don't need to assume any mathematical relationships about the forces. Just identify the forces and their directions. Now, a rule of thumb is that, unless there are electric or magnetic forces in your problem, gravity is the only one that acts without touching the object. All other forces are contact forces.

For the object hanging from a string, then, you have gravity acting downward. The only thing the object is in contact with is the string, so that is the second force - however, we call the force of a string "tension" instead of normal force, because it pulls, while a normal force pushes away. Since the string is directed upward, so is the tension.

So that's your free-body diagram: gravity down and tension up. But generally the only equation you can apply to the forces is Newton's second law.

$Sigmaoverrightarrow F = moverrightarrow a$

If we take the vertical component of this equation, make upward positive, and plug in the forces from the diagram, we have:

$T - mg = ma$

But if the object is hanging, it is not moving, so acceleration is zero:

$T - mg = 0T = mg$

So tension = gravitational force, as you said. But it is a result of Newton's second law with no acceleration - it is not something you could have known from the beginning, unless you have solved this problem before.