Environmental sound recognition (ESR) is an emerging research topic in audio pattern recognition. Many tasks are presented to resort to computational models for ESR in real-life applications. However, current models are usually designed for individual tasks, and are not robust and applicable to other tasks. Cross-task models, which promote unified knowledge modeling across various tasks, have not been thoroughly investigated. In this article, we propose a cross-task model for three different tasks of ESR: 1) acoustic scene classification; 2) urban sound tagging; and 3) anomalous sound detection. An architecture named SE-Trans is presented that uses attention mechanism-based Squeeze-and-Excitation and Transformer encoder modules to learn the channelwise relationship and temporal dependencies of the acoustic features. FMix is employed as the data augmentation method that improves the performance of ESR. Evaluations for the three tasks are conducted on the recent databases of detection and classification of acoustic scenes and event challenges. The experimental results show that the proposed cross-task model achieves state-of-the-art performance on all tasks. Further analysis demonstrates that the proposed cross-task model can effectively utilize acoustic knowledge across different ESR tasks.