Does one neurotransmitter travel all the way through the nervous system?
You've got a few things mixed up here:

A signal may travel down a pathway, passing through several neurons (e.g. around 4-8). However, this signal is not always in the same form: From one end of a neuron to the other, it is merely an electric potential which travels down the neuron's cell membrane (from the dendrite to the axon hillock, where the cell decides whether to carry the signal on; if yes, a new potential is sparked to travel down the axon).

When the potential reaches the end of a neuron, i.e. an axon terminal, it causes exocytosis of neurotransmitter. The signal has become chemical now. This diffuses through the synaptic cleft to the next neuron, where it acts on ion channels in the membrane to cause a new electric potential.

So the actual distance a neurotransmitter travels is only the distance between the two membranes at the synapse, i.e. the synaptic cleft. This is only about 20-40 nm wide, so it's fairly rapid :)