To fly, a creature will need to have the skill Flight. It is an attack skill and will only be active when triggered. When triggered, Flight will come into effect after the current assault, keep you flying until the next assault, and you will go back to the ground after the next assault (unless you triggered the Flight skill again). This is a short summary of how it happens: Flight triggered => attack from the ground => start flying => next turn => attack while flying => go back to the ground.
Flying creatures can not be hit from the ground, unless the attacker has one of the following skills. The skills needed to attack a flying creature are all found in the Air skill tree, and they are Jump and Master Leviticus.
It is possible to attack a flying enemy while you are flying. If your Dinoz are currently flying and trigger its multi-attack effect while attacking a flying enemy, only the first attack will successfully hit the target. The second attack will be counted as a ground attack.
Some skills also damage flying Dinoz:
There are only three Dinoz capable of flying, namely Cloudoz, Glidwings, and Pteroz. These Dinoz will have four level 1 Air skills available where all the others will only have 3. There are, however, no additional skills to be unlocked beyond Flight.