Right now I just play each string one by one and not 2 strings at once or anything
Does it harder and harder everytime I make progress with private guitar lessons?
No, but as you conquer one objective you need to move to another and increase, and hone your skills as you go. I've been playing awhile and am considered advanced, I would be bored with a beginning lesson, on the other hand, you would be overwhelmed trying some of my lessons.