Where can I get a SIM card at the Hyderabad (Telangana, India) airport, or somewhere on the way to Banjara Hills?

Pre-activated SIM cards are not sold in India. This means you have to buy a SIM card and then wait for some time for the number to get activated (this can vary from 2 to 24 hours). To buy a SIM card, you can visit official outlets of Indian telecom operators - Vodafone, Airtel, BSNL or Reliance Jio. There are a number of such stores in Hyderabad. A simple Google search for "Vodafone store" or "Airtel store" in an area will show you your closest one.

To buy a SIM as a foreign national you need to submit some documents. This article has a nice overview of the process. From the article:

If you want a prepaid (pay-as-you-go) connection, visit the nearest Vodafone, Airtel, or Reliance Jio store in India with the following documents.

• Passport, with a photocopy
• Photocopy of the visa
• Passport-sized photographs (usually just one needed, but carry four just in case since telecom operators require four photographs in some Indian states)

The executive at the stores will guide you about the exact documents but I feel you may need to submit proof of local address (like a hotel reservation) in addition to ones mentioned above.

Also remember that your existing device may not work for frequencies at which Indian operators work. You can find complete frequency bands here.