Do you always see the same side of the moon from the earth?

The simple answer is Yes. The reason is that the moon rotates on its axis as it goes around the Earth. The two rotations are exactly in time with each other so that the moon always keeps the same face towards the Earth.

The slightly more complicated answer is that we see slightly more than half of the moon. It turns on its axis at a constant speed, but it goes around the Earth in an elliptical path, so it goes very slightly faster at some points along its orbit than at others. The end result is that it appears to wobble from side to side slightly, showing a very slight amount more at the left at some times and at the right at others. So we can over the course of a month see a bit more than half the moon