How does the flow of ions along voltage-gated channels lead to nerve conduction along axons?
The flow of current during action potential generation is perpendicular to the axon length. The channels that transmit ions in neurons are often voltage-gated. Voltage-gated Na+ channels (VGSCs) open when the membrane depolarizes. So when excitatory neurotranmitters (e.g., glutamate) open up channels (e.g., AMPA receptors) in the dendritic region, anions (e.g., Na+) flow in. The ensuing membrane depolarization opens up VGSCs in the axon hillock, that subsequently opens VGSCs adjacently in the axon, that opens up VGSCs a little further up in the axon etc etc. Hence, while the flow of Na+ is perpendicular to the axonal membrane, the net effect is a wave of VGSCs opening up along the axon length. With some delay, voltage-gated K+ channels open that allow K+ to flow in, repolarizing the membrane, preparing the neuron for another action potential (fig. 1).


Fig. 1. Action potential mediated by step wise activation of voltage-gated ion channels. Source: Human Medical Physiology 

Hence, although neuronal ion fluxes and networks can be modeled by electric currents flowing in electronic circuitry (Fig. 2), nerve conduction is not anything like it. Current flow in electronic circuits is a flow of electrons. In action potentials in neurons it is ions that carry the transported charge.    

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Fig. 2. Hodgkin &amp; Huxley's axon model. The power sources are the electric gradients of Na+, K+ and a leak current source. The conductances are represented by those same ions. The cell membrane is modeled by a capacitor. Source: Bonabi et al. (2014)

Reference
- Bonabi et al., Front Neurosci (2014)