FAA Order JO 7110.65W - ATC Procedures, Chapter 10 outlines how ATC should handle procedures for emergency situations.
In the very first section though, it says:
Because of the infinite variety of possible emergency situations, specific procedures cannot be prescribed. However, when you believe an emergency
exists or is imminent, select and pursue a course of action which appears to be most appropriate under the circumstances and which most nearly conforms to the instructions in this manual.
So, for Mayday and Pan-Pan, it says in 10-1-1(a):
A pilot who encounters a Distress condition should declare an emergency by beginning the initial communication with the word “Mayday,” preferably
repeated three times. For an Urgency condition, the word “Pan-Pan” should be used in the same manner.
The use of Mayday and Pan-Pan declares an emergency, but doesn't tell ATC about the nature of that emergency. Therefore just saying those words doesn't trigger any particular method for handling the emergency other than probably the controller offloading other traffic and giving you priority handling. It'll probably also trigger a clearing of the airspace around you, and if you are on final or near an airport, a ground hold.
Provide priority to civilian air ambulance flights (call sign “MEDEVAC”). Use of the MEDEVAC call sign indicates that operational
priority is requested. When verbally requested, provide priority to AIR EVAC, HOSP, and scheduled air carrier/air taxi flights. Assist the pilots of
MEDEVAC, AIR EVAC, and HOSP aircraft to avoid areas of significant weather and turbulent conditions. When requested by a pilot, provide notifications to
expedite ground handling of patients, vital organs, or urgently needed medical materials.
It is recognized that heavy traffic flow may affect the controller’s ability to provide priority handling. However, without compromising safety, good judgment must be used in each situation to facilitate the most expeditious
movement of a MEDEVAC aircraft.
For Minimum Fuel
If an aircraft declares a state of “minimum fuel,” inform any facility to whom control jurisdiction is transferred of the minimum fuel problem and be alert
for any occurrence which might delay the aircraft en route.
Use of the term “minimum fuel” indicates recognition by a pilot that his/her fuel supply has reached a state where, upon reaching destination, he/she cannot accept any undue delay. This is not an emergency situation but merely an
advisory that indicates an emergency situation is possible should any undue delay occur. A minimum fuel advisory does not imply a need for traffic priority. Common sense and good judgment will determine the extent of assistance to be given in minimum fuel situations. If, at any time, the remaining usable fuel supply suggests the need for traffic priority to ensure a safe landing, the pilot should declare an emergency and report fuel remaining in minutes.
Really you should read through the entire (long) document to see how ATC is supposed to behave for certain situations. Chapter 2, Section 1 has a lot of "emergency" situations, but most of it boils down to giving priority and assistance that the pilot asks for.