Learn Turkish: Doktor her sabah tansiyonumu ölçmemi önerdi. - The doctor advised me to measure my blood pressure every morning.

This uses the indirect command structure you are currently learning. We take the verb root ölç- (measure) and add the short verbal noun ending -me, followed by the possessive -m (my). This creates "my measuring". Finally, we add the accusative case -i because the entire action is the direct object of the main verb önerdi (suggested). Literally, the doctor "suggested my measuring".

Why do both tansiyonumu and ölçmemi end with an accusative case marker?

They are direct objects belonging to two different verbs in the sentence. tansiyonumu has the accusative -u because it is the specific thing being measured (the object of ölçmek). ölçmemi gets the accusative -i because it is the specific action being suggested (the object of the main verb önermek).

Could I just say tansiyon ölçmemi without the possessive -um on blood pressure?

You could, but it changes the meaning slightly. Tansiyon ölçmemi sounds like the general activity of measuring blood pressure (for instance, if you were a nurse in training). By adding -um to make tansiyonumu, it specifically means the doctor wants you to measure your own personal blood pressure.

Would the grammar change if the doctor told me to measure it, rather than just advised it?