Being prior Air Force I have a literal out of the book answer for you. If anything seems confusing just ask, I can elaborate.
So I will do this in two parts, (after some comments) first the basics, these are the general principles by which Air power should be used, and second creatures (like varying aircraft) have different strengths and weaknesses so we can go into that.
Air Forces of any flavor should follow these basic rules/guides. What type of Air Power is being considered is not particularly relevant as all Air Power shares some of the same characteristics, mainly speed, maneuverability and the ability to strike basically anywhere.
Allow me to introduce you to AFDD1 (Air Force Doctrine Document 1)
Included in chapter three of the doctrine are the Tenets of Air and Space Power which are derived from the principles of war. (Unity of command, Objective, Offensive, Mass, Maneuver, Economy of Force, Security, Surprise, and Simplicity)
Tenets of Air and Space Power
- Centralized Control and Decentralized Execution
Centralized control of air and space power is the planning, direction,
prioritization, synchronization, integration, and deconfliction of air and space
capabilities to achieve the objectives of the joint force commander. Centralized
control of air and space power should be accomplished by an airman at the air component commander level who maintains a broad theater perspective in prioritizing the use of limited air and space assets to attain established objectives in any contingency across the range of operations. Centralized control maximizes the flexibility and effectiveness of air and
space power; however, it must not become a recipe for micromanagement, stifling the initiative subordinates need to deal with
combat’s inevitable uncertainties.
Decentralized execution of air and space power is the delegation of execution authority to responsible and capable lower level commanders to achieve effective span of control and to foster disciplined initiative, situational responsiveness, and tactical flexibility. It allows subordinates to exploit opportunities in rapidly changing, fluid situations. The benefits inherent
in decentralized execution, however, are maximized only when a commander clearly
communicates his intent
- Flexibility and Versatility
Air and space power is flexible and versatile. Although often used
interchangeably, flexibility and versatility are different. Flexibility allows air and space forces to exploit mass and maneuver simultaneously. Flexibility allows air and space operations to shift from one campaign objective to another, quickly and decisively; to “go downtown” on one sortie, then hit
fielded enemy forces the next; to rerole assets quickly from a preplanned
mission to support an unanticipated need for close air support of friendly
troops in contact with enemy forces. Versatility is the ability to employ air and space power effectively at the strategic, operational, and tactical levels of warfare. Air and space forces, unlike other military forces, have the
potential to achieve this unmatched synergy through asymmetric and parallel
Air and space power produces synergistic effects. The proper
application of a coordinated force can produce effects that exceed the
contributions of forces employed individually. The destruction of a large
number of targets through attrition warfare is rarely the key objective in
modern war. Instead, the objective is the precise, coordinated application
of the various elements of air, space, and surface power to bring disproportionate
pressure on enemy leaders to comply with our national will.
Air and space power offers a unique form of persistence. Air,
space, and information operations may be conducted continuously against a
broad spectrum of targets. Air and space power’s exceptional speed and
range allow its forces to visit and revisit wide ranges of targets nearly at
will. Air and space power does not have to occupy terrain or remain constantly
in proximity to areas of operation to bring force upon targets
Air and space power must achieve concentration of purpose. The
versatility of air and space power makes it an attractive option for almost
every combat task. Airmen must guard against the inadvertent dilution
of air and space power effects resulting from high demand. One of
the most constant and important trends throughout military history has
been the effort to concentrate overwhelming power at the decisive
time and place.
Air and space power must be prioritized. Demands for air and
space forces (because of their flexibility and versatility) will likely overwhelm air commanders in future conflicts unless appropriate priorities are
Air and space operations must be balanced. Balance is an essential
guideline for air commanders. Much of the skill of an air commander is
reflected in the dynamic and correct balancing of the principles of war and
the tenets of air and space power to bring air and space power together to
produce a synergistic effect.
Rather than go creature by creature I am going to list off some standard categories of air power usage and then we can slot creatures into each category.
Airlift/Search and Rescue: Airlift is simply the movement of people and cargo. There aren't a lot of creatures that would fit in this category, even with modern technology air transport is incredibly energy intensive and there just aren't a lot of creatures that have the size/cargo space.
The exception here would be support for a covert operations (basically the roles for which we use non-attack helicopters), dropping operatives off, or for search and rescue. Dragons in most versions can carry several people but considering the other abilities of dragons it seems like a misappropriation of assets when they could to more good elsewhere.
The Pegasus would fall into this category, essentially, like the helicopter it can be used to get troops to places they otherwise may not be able to reach. Hippogryphs and Griffons would be useful here as well.
This is where a medieval force will be best applied. Dragons could wipe out legions without even landing. Keep in mind that in warfare, the biggest bang for the buck is not always (nor even often) a military target. Strategic objectives, i.e. command and control, transportation, communications, supply chain etc. You can to a lot more damage to an army if you destroy their food supplies than if you attack them head on.
Stymphalian birds are also a great choice...if you can control them. They are not generally looked at as intelligent so much as a force of nature like a plague of locusts.
This is a weird one as we tend to think of fighters with missiles...dragons naturally come to mind, but compared to smaller aerial creatures the are big and slow a pack of giant eagles could shred a dragon's wings without much effort.
Giant birds thus fit well, giant eagles, roc's (in some uses they can breathe lightning which would be awesome), the phoenix (my personal favorite), wyverns (usually depicted as smaller than dragons) I will add as well, though not really birds.
This is where it can get fun in my opinion. Gargoyles obviously come to mind, hide as stone, strike while everyone is asleep. The cockatrice works as well, can turn people to stone...
Support Roles: Now, any good air force member will tell you that air power is not best utilized in support roles because it violates several tenets, offense most particularly. That said some mythological creatures are totally functionally different. Sylphs and Fairies can heal in some mythologies so that would be helpful.