If one adopt adopts a higher order attitude towards PA, then one makes the transition from reasoning within PA to wanting to understand the structure of all the models of PA, of which there is a rich diversity. This move is analogous to the common change in viewpoint one undergoes in abstract algebra, moving at first from studying an individual group or ring from inside, using the axioms directly, to studying structure theorems about all groups or all rings from the outside, and exploring especially how they relate to each other.
Indeed, many of the most fascinating aspects of PA as a first order theory, such as the independence results mentioned in the other answers, flow from the fact that there are so many different models of PA. Perhaps the most authoritive book on the structure of the models of Peano Arithmetic is:
This book investigates the collection of all models of PA, analyzing their substructure lattices, saturation properties, various kinds of extensions, cuts, automorphisms, order types and so on. I highly recommend it. It appears to be aimed at students who have had already had some exposure to basic mathematical logic, and for this reason may be more advanced than you wanted.
