In this paper, we propose to estimate the spatial curvature of the universe and the cosmic opacity in a model-independent way with expansion rate measurements, H(z), and type Ia supernova (SNe Ia). On the one hand, using a nonparametric smoothing method Gaussian process, we reconstruct a function H(z) from opacity-free expansion rate measurements. Then, we integrate the H(z) to obtain distance modulus mu(H), which is dependent on the cosmic curvature. On the other hand, distances of SNe Ia can be determined by their photometric observations and thus are opacity-dependent. In our analysis, by confronting distance moduli mu(H) with those obtained from SNe Ia, we achieve estimations for both the spatial curvature and the cosmic opacity without any assumptions for the cosmological model. Here, it should be noted that light curve fitting parameters, accounting for the distance estimation of SNe Ia, are determined in a global fit together with the cosmic opacity and spatial curvature to get rid of the dependence of these parameters on cosmology. In addition, we also investigate whether the inclusion of different priors for the present expansion rate (H-0: global estimation, 67.74 +/- 0.46 km s(-1) Mpc(-1), and local measurement, 73.24 +/- 1.74 km s(-1) Mpc(-1)) exert influence on the reconstructed H(z) and the following estimations of the spatial curvature and cosmic opacity. Results show that, in general, a spatially flat and transparent universe is preferred by the observations. Moreover, it is suggested that priors for H-0 matter a lot. Finally, we find that there is a strong degeneracy between the curvature and the opacity.