In this paper, we consider an unmanned aerial vehicle (UAV)-enabled radio access network (RAN) with the UAV acting as an aerial platform to communicate with a set of ground users (GUs) in a variety of modes of practical interest, including data collection in the uplink, data transmission in the downlink, and data relaying between GUs involving both the uplink and downlink. Under this general framework, two UAV operation scenarios are considered: periodic operation, where the UAV serves the GUs in a periodic manner by following a certain trajectory repeatedly, and one-time operation where the UAV serves the GUs with one single fly and then leaves for another mission. In each scenario, we aim to minimize the UAV periodic flight duration or mission completion time, while satisfying the target rate requirement of each GU via a joint UAV trajectory and communication resource allocation design approach. Iterative algorithms are proposed to find efficient locally optimal solutions by utilizing successive convex optimization and block coordinate descent techniques. Moreover, as the quality of the solutions obtained by the proposed algorithms critically depends on the initial UAV trajectory adopted, we propose new methods to design the initial trajectories for both operation scenarios by leveraging the existing results for solving the classic Traveling Salesman Problem (TSP) and Pickup-and-Delivery Problem (PDP). Numerical results show that the proposed trajectory initialization designs lead to significant performance gains compared to the benchmark initialization based on circular trajectory.