The Palomar Transient Factory (PTF) is an optical wide-field variability survey carried out using a camera with a 7.8 deg$^2$ field of view mounted on the 48 inch Oschin Schmidt telescope at Palomar Observatory. One of the key goals of this survey is to conduct high-cadence monitoring of the sky in order to detect optical transient sources shortly after they occur. Here, we describe the real-time capabilities of the PTF and our related rapid multiwavelength follow-up programs, extending from the radio to the γ-ray bands. We present as a case study observations of the optical transient PTF10vdl (SN 2010id), revealed to be a very young core-collapse (Type II-P) supernova having a remarkably low luminosity. Our results demonstrate that the PTF now provides for optical transients the real-time discovery and rapid-response follow-up capabilities previously reserved only for high-energy transients like gamma-ray bursts.