Short-term residential load forecasting (STRLF) holds great significance for the stable and economic operation of distributed power systems. Different households in the same region may exhibit similar consumption patterns owing to the analogous environmental parameters. Incorporating the spatiotemporal correlations can enhance the load forecasting performance of individual households. To this end, a spatiotemporal graph attention (STGA)-enabled Transformer is proposed for multivariate, multi-step residential load forecasting in this paper. Specifically, the multiple residential loads are cast to a graph and a Transformer with a graph sequence-to-sequence (Seq2Seq) structure is employed to model the multi-step load forecasting problem. Gated fusion-based STGA blocks are embedded in the encoder and decoder of the Transformer to extract dynamic spatial correlations and non-linear temporal patterns among multiple residential loads. A transform attention block is further designed to transfer historical graph observations into future graph predictions and alleviate the error propagation between the encoder and decoder. The embedding of multiple attention modules in the Seq2Seq framework allows us to capture the spatiotemporal correlations between residents and achieve confident inference of load values several steps ahead. Numerical simulations on residential data from three different regions demonstrate that the developed Transformer method improves multi-step load forecasting by 14.7% at least, compared to the state-of-the-art benchmarks.