Upon the arrival of emerging devices, including Extended Reality (XR) and Unmanned Aerial Vehicles (UAVs), the traditional bit-oriented communication framework is approaching Shannon's physical capacity limit and fails to guarantee the massive amount of transmission within latency requirements. By jointly exploiting the context of data and its importance to the task, an emerging communication paradigm shift to semantic level and effectiveness level is envisioned to be a key revolution in Sixth Generation (6G) networks. However, an explicit and systematic communication framework incorporating both semantic level and effectiveness level has not been proposed yet. In this article, we propose a generic task-oriented and semantics-aware (TOSA) communication framework for various tasks with diverse data types, which incorporates both semantic level information and effectiveness level performance metrics. We first analyze the unique characteristics of all data types, and summarise the semantic information, along with corresponding extraction methods. We then propose a detailed TOSA communication framework for different time critical and non-critical tasks. In the TOSA framework, we present the TOSA information, extraction methods, recovery methods, and effectiveness level performance metrics. Last but not least, we present a TOSA framework tailored for Virtual Reality (VR) data with interactive VR tasks to validate the effectiveness of the proposed TOSA communication framework.