In the last few decades, the forecast of climate and meteorological variables has continued to attract scientists’ attention globally since it is highly related to global warming. Climate forecasting is an important, yet often difficult task. In this paper, we explore the suitability of several time series methods for temperature and rainfall forecasting in Brazil. Autoregressive integrated moving averages (ARIMA) and exponential smoothing models are among the most popular models in time series forecasting, which have been widely applied to climate data during the past few decades. Artificial neural networks (ANNs) are flexible computing frameworks that have been applied to a wide range of forecasting problems with high accuracy. However, using ANNs to model climate problems has yielded mixed results in the literature. We investigate the predictive performance of several exponential smoothing models and a hybrid ANN algorithm using a case study of monthly temperature and rainfall data of Brazil for 30 years (1991–2020) with a view of predicting the change-points in climate fluctuations. Results from the Mann-Kendall trend analysis revealed the presence of seasonal trends in Brazil’s temperature and rainfall data over the years. The ANN algorithm is shown to outperform the traditional time series model of SARIMA and advanced exponential smoothing models in predicting Brazilian climate series. Results from this study would be useful for policymakers as well as climate modellers in making decisions about selecting the best models for forecasting climate time series.