What part of שחרית would you identify as "prayer"?
When the g'mara used the term "shacharis" it referred to Sh'moneh Esre, all of which is prayer. Likewise, when it used the term t'fila, generally translated as "prayer", it meant Sh'moneh Esre alone. It is a 3- (or 4-, including supplemental petitions at the end) part prayer, beginning with praise, continuing with requests, and concluding with gratitude.
If you are referring to the entirety of the recitation in the morning, from Modeh Ani through Alenu L'shabe'ach, and using a broader definition of "prayer" than above, then the question may be more easily addressed by asking which parts are not prayer. Everything that contains one of the aforementioned elements of praise, request, or gratitude can rightly be considered prayer, which leaves a small number of obligatory blessings at the beginning, and the recital of Sh'ma.