The first choice is a red herring - it isn't really a choice at all - it's just a bit of TV showmanship. The first choice doesn't matter, because it's not acted on. You may have a 1 in 3 chance of selecting the right door, but your chance of winning is nil. Whichever door you choose, the host will open a different one.

The only choice which matters is the second choice, after one of the doors has been taken out of contention so it's then a straight 50/50 chance between 2 doors, one of which must contain the car and the other the goat. It's an entirely new choice, which may or may not be the same as the first choice. The first choice doesn't have a bearing on it, it only appears to.
Yes, given a straight choice between two doors, where your only knowledge is that one hides a goat, the other a car, your chance is 50/50.

But...

This is not a simple choice. You have already selected a door, and are being asked to choose whether to change. This means that the first choice affects the outcome. The door you have already chosen was a choice from 3, where you knew that it was more likely (2/3) that you would have chosen a door with a goat behind.
Someone just seeing the two doors and being offered the choice does not have the additional knowledge, so has a 50% chance of finding the car. With your additional knowledge (by swapping) your chance increses to 66%+.

No wonder it confuses the experts!

Cheers
Nigel (recreational mathematician)