Spiking neural networks (SNNs) are well-suited to process asynchronous event-based data. Most of the existing SNNs use rate-coding schemes that focus on firing rate (FR), and so they generally ignore the spike timing in events. On the contrary, methods based on temporal coding, particularly time-to-first-spike (TTFS) coding, can be accurate and efficient but they are difficult to train. Currently, there is limited research on applying TTFS coding to real events, since traditional TTFS-based methods impose one-spike constraint, which is not realistic for event-based data. In this study, we present a novel decision-making strategy based on first-spike (FS) coding that encodes FS timings of the output neurons to investigate the role of the first-spike timing in classifying real-world event sequences with complex temporal structures. To achieve FS coding, we propose a novel surrogate gradient learning method for discrete spike trains. In the forward pass, output spikes are encoded into discrete times to generate FS times. In the backpropagation, we develop an error assignment method that propagates error from FS times to spikes through a Gaussian window, and then supervised learning for spikes is implemented through a surrogate gradient approach. Additional strategies are introduced to facilitate the training of FS timings, such as adding empty sequences and employing different parameters for different layers. We make a comprehensive comparison between FS and FR coding in the experiments. Our results show that FS coding achieves comparable accuracy to FR coding while leading to superior energy efficiency and distinct neuronal dynamics on data sequences with very rich temporal structures. Additionally, a longer time delay in the first spike leads to higher accuracy, indicating important information is encoded in the timing of the first spike.