Third-generation neural networks, or Spiking Neural Networks (SNNs), aim at harnessing the energy efficiency of spike-domain processing by building on computing elements that operate on, and exchange, spikes. In this paper, the problem of training a two-layer SNN is studied for the purpose of classification, under a Generalized Linear Model (GLM) probabilistic neural model that was previously considered within the computational neuroscience literature. Conventional classification rules for SNNs operate offline based on the number of output spikes at each output neuron. In contrast, a novel training method is proposed here for a first-to-spike decoding rule, whereby the SNN can perform an early classification decision once spike firing is detected at an output neuron. Numerical results bring insights into the optimal parameter selection for the GLM neuron and on the accuracy-complexity trade-off performance of conventional and first-to-spike decoding.