How far from the earth do you have to be to start falling toward the moon?

Factual Questions
1

Is there some point in space between the Earth and the Moon where the latter’s gravity becomes predominate and you start falling toward it rather than the Earth? Is there point where the Sun’s gravity is stronger than the two? Let’s assume that a mass at that point has an initial velocity of 0.

Thanks,
Rob

2

Seems easiest to just cite this:

3

This might also be useful.