How does inertia relate to the flight of an airplane?

This question isn't very well-defined--inertia relates to the flight of an airplane in different ways, depending on which component of the flight is being examined.

Inertia is a quality related to an object's resistance to a change in its motion. The most common physical quantity related to inertia is an object's mass. The "Law of Inertia", which is often stated as (though Galileo had already stated it, and others had stated it before Galileo), states that objects tend to maintain constant velocity unless acted upon by an unbalanced force.

When there IS an unbalanced force, tells you what happens: the rate at which an object's momentum changes is equal to the force. Assuming constant mass an non-relativistic velocities (i.e. assuming that the velocity is less than about 10% of the speed of light, which it is for airplanes!), you can say that an object's is equal to the force applied to it divided by the object's mass.

When the airplane is grounded, the jets are required to apply a force in the horizontal direction to accelerate the plane (it started from rest, and its tendency is to remain at rest--you feel like you want to go backward because at each instant, you were previously at a lessor velocity).

After the plane has reached a great enough horizontal speed, the vertical lift force created by the interaction between the plane's wings with the air moving past eventually balances out and then becomes greater than the force due to gravity keeping the plane on the ground, thus resulting in a net vertical force upward. This lifts the plane off the ground.

After the plane has reached cruising altitude at constant velocity, passengers feel normal, as if they are motionless on land, because there is no net force acting on either them or the plane. Outside the plane, the air resistance is balanced by the thrust in the horizontal direction, and lift is balanced by gravity in the vertical direction.

Finally upon landing, the jets are turned down so that air resistance provides a greater force against the plane's motion than the jet thrust provides in the direction of the motion. The result is a slowing of the plane. With less horizontal velocity, there is also less lift, so the plane descends until it lands.