So how is flight actually achieved? In an airplane, lift is created by the wings traveling through a fluid (hopefully air), but in something a bit less elegant like a rocket or jetpack lift is generated by shooting stuff toward the earth really fast. The "stuff" is usually some extremely hot gas, maybe steam or exhaust vapor, that is ejected downward (read: against gravity) at ridiculous speeds. In general, if you want to fly without wings you need to generate enough thrust to counteract gravity.
Thrust is calculated by taking the product of two values: the rate of ejection of mass (the "stuff"), and the speed of the mass being ejected.
where T is Thrust, v is the velocity of the mass being ejected, and dm/dt is the amount of mass being ejected over time. So, we need to know these two values.
Water has a mass density of 1000 kg/m^3, air a density of 1.225 kg/m^3, and fire is massless (it's more of an expression of energy than an actual "thing"). So if mass-ejection rate were held equal for both water and air, then water would provide the most thrust because it is the most massive.
But that's only half the equation. The speed of ejection of the propellant is just as important as the mass of the propellant. In theory, if we could expel air 1000x faster than water, air would be just as good a candidate for propellant as water. In practice, however, if you can manage to spew air out a nozzle at extreme speed, you will also be able to manage that same speed with water. We're forgetting our massless third element, though.
By itself, fire is useless as a propellant because it isn't an actual thing. As an expression of energy combined with air or water, however, we get to go very interesting places. Using fire to heat the propellant at high speeds is easily the most effective method of thrust. Many rockets use a fuel of Liquid Oxygen and Liquid Hydrogen, which when burned produce water. The extremely hot water (technically steam) is ejected out the nozzle at extreme speeds by its own pressure, and produces thrust. Jet engines, on the other hand, produce thrust by burning fuel and heating air. The heated air is ejected out the rear of the jet at high speeds and produces thrust.
So what about our aeronautic castor? If he can cast only a single element at a time, his best bet will be water since it is the most massive option and thus able to project the most thrust as he casts it. If he can cast multiple elements, he could either use fire + water creating a steam rocket (like many modern rockets) or use fire + air to create an air-breathing jet. It is also possible that some combination of the three might be even more efficient if his powers limit the amount of mass he can cast at once (though that will require some discussion of specific impulse and some other more complicated subjects).
I hope that helps!