The force required to lift an object with weight mg must be slightly greater than mg to produce a net force upwards. There's one thing I'm struggling to understand with this concept though and would appreciate if someone can clarify it for me. If an object is resting on a table then it will have a reaction force (the normal force) also acting upwards on the object. Since the net force on this object is zero as the normal/weight forces cancel out, then shouldn't any force applied upwards to the object cause it to start moving, regardless of whether the force's magnitude is less than mg or not?
The normal force only acts on that object while that object is in contact with the surface it is resting on. It is also proportional to the force being applied onto the surface - which is not necessarily $mg$ but would be if there were no other forces involved. Say you went to lift the object off that surface, and you applied a force $F<mg$ upwards. What you would get is a gravitational force going down of $mg$ and a normal force of $N=mg-F$ going up and your force of $F$ going up. In the end, the situation is still balanced and the object doesn't move. The normal force does not always provide a $mg$ force going up. It's only enough to keep the object from moving down through the surface.
In a sense, you are correct.
Consider the mass resting not on a table, but on a very easily stretched trampoline.
The mass is at rest; the force of gravity is exactly balanced by the force exerted by the elastic stretching of the trampoline.
If you apply the slightest upward force, the mass will begin to accelerate upward. However, this allows the elastic trampoline to contract, so it exerts less upward force. The net upward force thus decreases, bringing the upward acceleration down to (and past) zero. The object eventually stops at a new level, with gravity balanced by the reduced upward force and the new upward force from you.
Now consider the table to be a really stiff trampoline...