Unmanned aerial vehicles (UAVs) have become important in many applications including last-mile deliveries, surveillance and monitoring, and wireless networks. This paper aims to design UAV trajectories that simultaneously perform multiple tasks. We aim to design UAV trajectories that minimize package delivery time, and at the same time provide uniform coverage over a neighborhood area which is needed for applications such as network coverage or surveillance. We first consider multi-task UAVs for a simplified scenario where the neighborhood area is a circular region with the post office located at its center and the houses are assumed to be uniformly distributed on the circle boundary. We propose a trajectory process such that if according to which the drones move, a uniform coverage can be achieved while the delivery efficiency is still preserved. We then consider a more practical scenario in which the delivery destinations are arbitrarily distributed in an arbitrarily-shaped region. We also do not assume any restrictions on the package arrivals. We show that simultaneous uniform coverage and efficient package delivery is possible for such realistic scenarios. This is shown using both rigorous analyses as well as simulations.