How did the Pilgrims adapt to the strategies of fighting used by the Indians?

One of the most basic methods was through the use of advanced technology. Natives had never seen such things as horses or guns, and had no prior experience with combating them. Pizarro, for example, famously used a cannon to disorient and distract an Inca army outnumbering his vastly, scattering the massive horde of opponents.
Horses confounded the Natives (at least until the Plains Indians acquired them), and could be used to scare and startle the Natives as well.
Indians were proud of honor, and tended to think the Europeans were as well. They would do things like honor battles, strapping themselves to the ground so they could not flee. Europeans would simply shoot them.