Many sensing systems remotely monitor/measure an environment at several sites, and then report these observations to a central site. We propose and investigate several practical algorithms for joint routing and compression of data files as they are forward from remote nodes to a central site, with the goal of minimizing the communication cost incurred. Our algorithms are practical in that they do not assume that nodes have a priori information about the correlation structure (and resulting compression gains) of the individual measurements at a given sensor or among multiple sensors. Instead, this correlation structure is learned as pieces of the files are routed and jointly compressed on their way to the sink, and routes are adaptively changed as the nodes learn more about the correlation structure of the data.