One of my PPL manuals had an illustration of a boat crossing the river. If it didn't point upriver to compensate for the current, it would end up downriver from it's objective, or If the boat crossed the river with the objective always kept at the 12 O' clock position, it's course ended up being a big arc.
Correct. Think of the wind as the entire atmosphere moving sideways along the surface of the earth. If you are on the ground, you feel the atmosphere moving by you as wind. But, if you are in the air, you simply move with the atmosphere as it slides across the ground. Discounting instabilities in the atmosphere (gusts, wind shear, thermals, etc.), the airplane flys exactly the same in calm winds as it does in a 50 mph wind because it is moving
with the airmass.
The path that you are taking through the airmass is defined by heading (where the nose of the airplane is pointing) and true airspeed. If the airmass is moving in relation to the surface of the earth ("wind"), then you are traveling with it. As I pointed out before, this doesn't change the way the plane flys, but what it does is changes the path it is taking along the ground. If you were to make an arrow showing your path through the airmass (heading & TAS) and add to it an arrow showing the movement of the airmass across the ground (wind) then the resulting arrow would give you the airplane's actual path over the ground. If you've ever used a manual E6B this is exactly what you are doing when you use the wind face. The direction of your path along the ground is your track and the speed at which you move along the ground is your ground speed.
So, if you are flying with a 20 knot tailwind, the aircraft would handle exactly the same as it would in calm winds, but your ground speed would be 20 knots faster than your true airspeed.
If you are flying with a 20 knot cross wind from left to right, the aircraft would handle exactly the same, but your track would be a few degrees to the right of its heading.
The challenge comes when you have to take the airplane which is moving with the airmass, and land it on a runway which is not. You can see now the reason pilot's always try to land into the wind. If you touch down at 55 knots true airspeed into a 10 knot headwind, then the aircraft is actually only moving forward at 45 knots in relation to the runway. And the tricky part is if you have a 10 knot crosswind, then the plane is trying to move sideways in relation to the runway at 10 knots. One of the harder parts of getting your pilot certificate is to learn how to deal with crosswind landings.