With the expansive deployment of ground base stations, low Earth orbit (LEO) satellites, and aerial platforms such as unmanned aerial vehicles (UAVs) and high altitude platforms (HAPs), the concept of space-air-ground integrated network (SAGIN) has emerged as a promising architecture for future 6G wireless systems. In general, SAGIN aims to amalgamate terrestrial nodes, aerial platforms, and satellites to enhance global coverage and ensure seamless connectivity. Moreover, beyond mere communication functionality, computing capability is increasingly recognized as a critical attribute of sixth generation (6G) networks. To address this, integrated communication and computing have recently been advocated as a viable approach. Additionally, to overcome the technical challenges of complicated systems such as high mobility, unbalanced traffics, limited resources, and various demands in communication and computing among different network segments, various solutions have been introduced recently. Consequently, this paper offers a comprehensive survey of the technological advances in communication and computing within SAGIN for 6G, including system architecture, network characteristics, general communication, and computing technologies. Subsequently, we summarize the pivotal technologies of SAGIN-enabled 6G, including the physical layer, medium access control (MAC) layer, and network layer. Finally, we explore the technical challenges and future trends in this field.