The SAGE-Spec Spitzer Legacy program is a spectroscopic follow-up to the SAGE-LMC photometric survey of the Large Magellanic Cloud carried out with the Spitzer Space Telescope. We present an overview of SAGE-Spec and some of its first results. The SAGE-Spec program aims to study the life cycle of gas and dust in the Large Magellanic Cloud and to provide information essential to the classification of the point sources observed in the earlier SAGE-LMC photometric survey. We acquired 224.6h of observations using the infrared spectrograph and the spectral energy distribution (SED) mode of the Multiband Imaging Photometer for Spitzer. The SAGE-Spec data, along with archival Spitzer spectroscopy of objects in the Large Magellanic Cloud, are reduced and delivered to the community. We discuss the observing strategy, the specific data-reduction pipelines applied, and the dissemination of data products to the scientific community. Initial science results include the first detection of an extragalactic 21um feature toward an evolved star and elucidation of the nature of disks around RV Tauri stars in the Large Magellanic Cloud. Toward some young stars, ice features are observed in absorption. We also serendipitously observed a background quasar, at a redshift of z~~0.14, which appears to be hostless.