Flow in porous media is known to be largely affected by pore morphology. In this work, we investigate the effects of pore geometry on the transport and spatial correlations of flow through porous media in two distinct pore structures arising from three-dimensional assemblies of overlapping and nonoverlapping spheres. Using high-resolution direct numerical simulations (DNS), we perform Eulerian and Lagrangian analysis of the flow and transport characteristics in porous media. We show that the Eulerian velocity distributions change from nearly exponential to Gaussian distributions as porosity increases. A stretched exponential distribution can be used to represent this behavior for a wide range of porosities. Evolution of Lagrangian velocities is studied for the uniform injection rule. Evaluation of tortuosity and trajectory length distributions of each porous medium shows that the model of overlapping spheres results in higher tortuosity and more skewed trajectory length distributions compared to the model of nonoverlapping spheres. Wider velocity distribution and higher tortuosity for overlapping spheres model give rise to non-Fickian transport while transport in nonoverlapping spheres model is found to be Fickian. Particularly, for overlapping spheres model our analysis of first-passage time distribution shows that the transport is very similar to those observed for sandstone. Finally, using three-dimensional (3D) velocity field obtained by DNS at the pore-scale, we quantitatively show that despite the randomness of pore-space, the spatially fluctuating velocity field and the 3D pore-space distribution are strongly correlated for a range of porous media from relatively homogeneous monodisperse sphere packs to Castlegate sandstone.