At high altitudes there's less oxygen in air so the first effect is hypoxia and decreased dissolved oxygen in the blood.
The immediate response will be stimulation of peripheral chemoreceptors which in turn in send signals to respiratory drives to breath more.
So hyperventilation develops which in turn leads to Respiratory Alkalosis.
Respiratory alkalosis will shift the hemoglobin dissociation curve to the left so that Hb can pick up O2 easier.
The kidney will respond to alkalosis by generating hydrogen ions and this will correct the PH back to normal.
After 2-3 days the 2,3 Diphosphoglycerate level in RBC will increase and this will shift the Hb curve (this time) to the right making O2 delivery easier to tissues. (Athletes go to mountains before the olympics, so that when they return to sea level, they have a right shifted Hb curve and so easier O2 delivery to their muscles during their competitions).
Another change that start right away is the increase of eryrthopoietin secretion but this takes a week or so before we can see polycythemia and increased hematocrit level. These changes in turn will correct the dropped down oxygen content of the blood.
Note that no matter how long you stay in high altitudes the PAO2, PACO2, and Hemoglobin saturation will remain decreased as we breath less FIO2 and we continue to hyperventilate.