Where in the neuron does the signal travel so it can later be transmitted to other neurons?

Neurons have 2 types of branches: dendrites and an axon
     
     dendrites receive information
     axon sends information

Dendrites receive information in the form of electrical potentials, some positive and some negative and at different rates. These signals are mixed and added together in the neuron, and if the sum at a point at the beginning of the axon (known as axon hillock) is high enough to reach a certain threshold, an electrical signal known as an action potential is fired. This signal is sent from the axon hillock, along the axon, to the end (known as the axon terminal), where it can communicate with other neurons.

So in summary, the signal travels:
dendrites -> axon hillock -> along the axon -> axon terminal -> synapse with other neurons

Note: the soma or cell body of a neuron (the part containing the nucleus) is located differently in different neurons (unipolar, bipolar, etc.). In the typical neuron, which is known as multipolar, the cell body is located between the dendrites and the axon hillock in this process.