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This paper considers fully discrete finite element approximations to subdiffusion equations with memory in a bounded convex polygonal domain. We first derive some regularity results for the solution with respect to both smooth and nonsmooth initial data in various Sobolev norms. These regularity estimates cover the cases when $u_0\in L^2(\varOmega )$ and the source function is Hölder continuous in time. The spatially discrete scheme is developed using piecewise linear and continuous finite elements, and optimal-order error bounds for both homogeneous and nonhomogeneous problems are established. The temporal discretization based on the L1 scheme is considered and analyzed. We prove optimal error estimates in time for both homogeneous and nonhomogeneous problems. Finally, numerical results are provided to support our theoretical analysis.