Given their extensive geographic coverage, low Earth orbit (LEO) satellites are envisioned to find their way into next-generation (6G) wireless communications. This paper explores space-air-ground integrated networks (SAGINs) leveraging LEOs to support terrestrial and non-terrestrial users. We first propose a practical satellite-ground channel model that incorporates five key aspects: 1) the small-scale fading characterized by the Shadowed-Rician distribution in terms of the Rician factor K, 2) the path loss effect of bending rays due to atmospheric refraction, 3) the molecular absorption modelled by the Beer-Lambert law, 4) the Doppler effects including the Earth's rotation, and 5) the impact of weather conditions according to the International Telecommunication Union Recommendations (ITU-R). Harnessing the proposed model, we analyze the long-term performance of the SAGIN considered. Explicitly, the closed-form expressions of both the outage probability and of the ergodic rates are derived. Additionally, the upper bounds of bit-error rates and of the Goodput are investigated. The numerical results yield the following insights: 1) The shadowing effect and the ratio between the line-of-sight and scattering components can be conveniently modeled by the factors of K and m in the proposed Shadowed-Rician small-scale fading model. 2) The atmospheric refraction has a modest effect on the path loss. 3) When calculating the transmission distance of waves, Earth's curvature and its geometric relationship with the satellites must be considered, particularly at small elevation angles. 3) High-frequency carriers suffer from substantial path loss, and 4) the Goodput metric is eminently suitable for characterizing the performance of different coding as well as modulation methods and of the estimation error of the Doppler effects.