This is somewhat of a trick question - technically lightning does both. Let's take a look at the process through which lightning is known to be formed. Lightning occurs because of a difference in charge between a storm cloud and the ground.

First, the base of a cloud sends down a little electric discharge, called a stepped leader. It descends to the ground in steps, each about 50 yards (about 46 meters) in length. This process is extremely fast and impossible to see with the naked eye. Each step is less than a millionth of a second long. The interval between steps works out to about fifty-millionths of a second. This process can only be observed with the assistance of extremely quick-exposure cameras.

The stepped leader generally moves at about 75 miles per second (120 km/s) towards the ground. A typical trip duration is 20 milliseconds. Atoms pass along electrical charge much more quickly than sound vibrations.

The stepped leader carries tons of negative charge. As it nears the ground, it induces enormous quantities of positive charge in the earth, especially at the tips of tall objects. Because opposites attract, the stepped leader and the negative charge at the ground reach towards each other and quickly meet. The path from storm cloud to the surface is complete and "the floodgates are open", so to speak.

Because the cloud is filled with negative charge, it has a lot of current to offer to the newly created discharge path. This charge quickly moves from being distributed throughout the cloud to being concentrated at the point where the stepped leader first dropped from the cloud, into the ground or an elevated object. This discharge is called the return stroke, and is what we think of when we hear the word "lightning".

The return stroke takes around 100 millionths of a second to reach the ground. The immense flash generated is enough to leave an afterimage in our eyes for seconds at a time, giving us the illusion that the lightning flash is longer than it really is. In reality, our eyes cannot resolve any of the steps involved. We only see the final product - a lightning bolt.

So, to answer the original question - put roughly, lightning does move from cloud to ground - if you consider only the return stroke. The whole process begins in the cloud, goes to the ground, then goes back to the cloud, then goes from the entire cloud to the ground. That's how lightning works.