There are certain missions that will allow you to perform a take-off sequence. This can be skipped though we don’t recommend it since performing this sequence will yield bonus points at the end of the mission. Depending on the plane you’re using, you can either take off from an airfield or from a carrier.
Taking off is very simple; just hold R2 down to accelerate. While on a runway, your aircraft will automatically lift from the ground and take off.
When launching from a carrier, you have to tilt your aircraft’s nose up a bit after getting catapulted off the carrier deck to avoid crashing to the sea. Keep holding R2 to maintain speed until you're completely cleared.