We can tell from two different lines of evidence—geodetic and geologic—that the lithospheric plates move. Even better, we can trace those movements back in geologic time.
Geodesy, the science of measuring the Earth's shape and positions on it, lets us measure plate motions directly using GPS, the Global Positioning System. This network of satellites is more stable than the Earth's surface, so when a whole continent moves somewhere at a few centimeters per year, GPS can tell. The longer we do this, the better the accuracy, and in much of the world the numbers are quite precise by now.
Another thing GPS can show us is tectonic movements within plates. One assumption behind plate tectonics is that the lithosphere is rigid, and indeed that is still a sound and useful assumption. But parts of the plates are soft in comparison, like the Tibetan Plateau and the western American mountain belts. GPS data helps us separate blocks that move independently, even if only a few millimeters per year. In the United States, the Sierra Nevada and Baja California microplates have been distinguished this way.
Three different geologic methods help determine the trajectories of plates: paleomagnetic, geometric and seismic. The paleomagnetic method is based on the Earth's magnetic field.
In every volcanic eruption, the iron-bearing minerals (mostly magnetite) become magnetized by the prevailing field as they cool. The direction they're magnetized in points to the nearest magnetic pole. Because oceanic lithosphere forms continuously by volcanism at spreading ridges, the whole oceanic plate bears a consistent magnetic signature. When the Earth's magnetic field reverses direction, as it does for reasons not fully understood, the new rock takes on the reversed signature. Thus most of the seafloor has a striped pattern of magnetizations, as if it were a piece of paper emerging from a fax machine (only it's symmetrical across the spreading center). The differences in magnetization are slight, but sensitive magnetometers on ships or aircraft can detect them.
The most recent magnetic-field reversal was 781,000 years ago, so mapping that reversal gives us a good idea of spreading speed in the most recent geologic past.
The geometric method gives us the spreading direction to go with the spreading speed. It's based on the transform faults along the mid-ocean ridges. If you look at a spreading ridge on a map, it has a stairstep pattern of segments at right angles. If the spreading segments are the treads, the transforms are the risers that connect them. Carefully measured, those transforms yield the spreading directions. With plate speeds and directions, we have velocities that can be plugged into equations. These velocities match the GPS measurements nicely.
Seismic methods use the focal mechanisms of earthquakes to detect the orientation of faults. Although less accurate than paleomagnetic mapping and geometry, they are useful in parts of the globe that aren't well mapped and have no GPS stations.
We can extend measurements into the geologic past in several ways. The simplest one is to extend paleomagnetic maps of the oceanic plates farther from the spreading centers. Magnetic maps of the seafloor translate precisely into age maps. The maps also reveal how the plates changed velocity as collisions jostled them into rearrangements.
Unfortunately the seafloor is relatively young, nowhere more than about 200 million years old, because eventually it disappears beneath other plates by subduction. As we look deeper into the past we must rely more and more on paleomagnetism in continental rocks. As plate movements have rotated the continents, the ancient rocks turned with them, and where their minerals once indicated north they now point somewhere else, toward "apparent poles." If you plot these apparent poles on a map, they appear to wander away from true north as rock ages go back in time. In fact, north does not change (usually), and the wandering paleopoles tell a story of wandering continents.
These two methods, seafloor magnetization and paleopoles, combine into an integrated timeline for the motions of the lithospheric plates, a tectonic travelogue that leads smoothly up to today's plate movements.