Routing in wireless ad hoc networks is challenging mainly due to unreliable wireless links/channels. Geographic opportunistic routing (GOR) was proposed to cope with the unreliable transmissions by exploiting the broadcast nature and spatial diversity of the wireless medium. Previous studies on GOR has focused on networks with a single channel rate. The capability of supporting multiple channel rates, which is common in wireless systems, has not been carefully studied for GOR. In this paper, we carry out a study on the impacts of multiple rates, as well as candidate selection, prioritization and coordination, on the throughput of GOR. We propose a new local metric, expected one-hop throughput (EOT), to characterize the trade-off between the packet advancement and medium time cost. We further propose a local rate adaptation and candidate selection algorithm to maximize this metric. Simulation results show that the multi-rate GOR (MGOR) incorporating the rate adaptation and candidate selection algorithm efficiently forwards the packet to the destination with higher throughput than the corresponding geographic routing and pure opportunistic routing operating at any single rate. EOT is shown to be a good local metric to achieve high path throughput for MGOR.