(The entire question assumes I am to the left of said object). If I'm pushing on an object on a horizontal plane to the right (in my physics class we're disregarding air resistance right now), I'm exerting a force on the object that is directed toward the right, and it is exerting a reactionary force of equal magnitude on me, directed toward the left. So, for the object to move, the force I have to exert on it has to have a greater magnitude than the horizontal friction force the ground exerts on the object (that friction force exerted to the left). My question is, if the object exerts a force directed to left on me, and the friction force exerted on me is to the left, then why do I not move to the left?
You do not feel the frictional force of the object independently, you only feel the reactionary force of the object (including the frictional force). The frictional force you feel is due to the interaction between your feet and the ground. Since you are pushing the object to the right, you must be pushing left against the ground, which means that the frictional force acting on you is directed to the right.