I just finished reading the first section of the Principia, on definitions. In discussions of Definitions 3 ("Inherent force of matter") and 4 ("Impressed force") Newton repeatedly refers to the "force of inertia." Given that a person pushing on an object to change its velocity feels a force from the object when they make contact (the normal force, equal and opposite to the impressed force) I suppose that one can forgive him for this conceptual error. I am amused by the thought of modern physics pedagogues (whether old school traditionalists lecturing kids about how inertia isn't a force or hip and modern interactive instructors giving kids group activities on the top) scolding him.
Then he talks about the difference between absolute motion and relative motion, absolute motion being motion relative to space and relative motion being relative to another object. He makes the important point that you cannot discern absolute motion by looking at the relative motion of two different objects because you don't know the absolute motion of either. However, he then explores his famous bucket thought experiment to argue that there are some cases in which absolute acceleration can be inferred. I've already discussed that thought experiment and Mach's reaction to it, so I have nothing to add here. I will just say that the real significance of the Principia is not his conceptual understanding of physics (a modern pedagogue could easily find much to scold him for) but rather his ability to add new ideas to a big, bold, framework and then apply those ideas unflinchingly and with no reluctance to generalize, and extract predictions for the motion of the planets.
Next up: His axioms, or laws of motion.