Can a plate on the lithoshpere move itself or do all of them have to move?
Tectonic plates move in relation to each other and the amount of spreading that is done under the ocean. The spreading is where the magma comes to the surface of the crust under the water and pushes the plates apart.
Not all plates move at the same speed.
If one side of the plate is being push, the other side of the plate must either subduct (go under the lighter plate) or they lock up and then the movement is upward, creating mountain ranges.
A good example of this would be the rocky mountains, as these were pushed upward due to tectonic plate movement.
When these tectonic plates move, they generate earthquakes. The also generate volcanoes as the material that is subducted generates heat and causes the rock to melt into magma and then that turns into lava when it comes to the surface and makes a flow of material.
When you look carefully at all the data, one can easily see that the plates are related to each other and the movement of one will cause a movement of another.
That is how continents have moved around the globe.
National Geographic had a video out on just tectonic plates called, The Ring of Fire. It provides a good look at how the plates move.
So to answer your question directly, no, one plate cannot move by itself without another plate moving.