Graphene is a promising two-dimensional material for photo-detection owing to its high mobility, broadband optical absorption, zero band gap nature, and tunable carrier concentration through electrical gating. Despite these unique properties, its 2.3% optical absorption from ultraviolet to infrared wavelengths and short carrier lifetime has limited its usage for practical applications. In this work, we present a broadband, high responsivity, and high speed graphene photodetector. By use of plasmonic nanoantennas, an incident optical field can be strongly concentrated in close proximity to the metallic nanoantennas. This significantly reduces the drift path length of the majority of photo-generated carriers in graphene to the plasmonic nanoantennas that serve as the photodetector contact electrodes. As a result, a large number of the photo-generated carriers can drift to the photodetector contact electrodes despite the short carrier lifetime of graphene, offering high responsivity levels. Moreover, the photodetector is designed to offer high speed operation by minimizing the capacitive parasitics induced by the plasmonic nanoantennas. We demonstrate a broadband photo-detection operation covering the wavelength regime from 800 nm to 1800 nm. We achieve responsivity levels as high as 0.6 A/W at 800 nm, which is close to the theoretical limit of 0.65 A/W. In summary, the combination of the high-responsivity, broad bandwidth, and high-speed performance of the presented plasmonics-enhanced graphene photodetector could find many applications in future optical communication, imaging and sensing systems.