To discuss tides, first we have to discuss what forces are acting on the oceans. If the moon was the only force pulling the oceans then we would expect high tides at exactly 12 midnight and 12 noon everyday (which is not the case). The gravity of the moon and sun draw the ocean towards their centers. The moon causes a bigger affect on ocean tides than the sun because although the sun is larger it's also much much further away. All of this means that high and low tides are determined mainly by a tug-of-war between the pull of gravities of the earth, the moon and the sun.The size of high and low tides are determined by how the pull of the moon and sun line up-- If the moon and sun are on the same side of the earth, they are both pulling the ocean in the same direction so you get a bigger tide (called a spring tide). If they are pulling in opposite directions then you get what's called a neap tide. So not all high tides are equal, and not all low tides are equal, depending on the position of the moon relative to the sun.

You are correct by saying that when you see the moon, it is pulling the ocean in front of you outwards from the earth, but how low or high of a tide also depends on where the sun is. Also, the moon takes longer to rotate around the earth than 24 hours, in fact it takes 24 hours and 50 minutes, so the tide cycle (high vs. low) changes every day.

Here is a great animation that shows how the tides change within a day, but that the cycle takes more than 24 hours. Click here to watch