This paper presents the Internet of Smart Grid Things (IoSGT) architecture, which is understood as an ecosystem of cutting-edge technologies that work together to enable advanced SG applications, which run at core/edge cloud datacenter premises connecting an underlying IoT networking infrastructure. The IoT-to-Edge-to-Cloud continuum paradigm that the IoSGT architecture yields aim to accelerate a plethora of Smart Grid (SG) new generation systems, which will advance current facilities at unprecedented levels. We materialized the proposed architecture in the form of a prototype atop a testbed entailing real-world technologies to lay high-accurate experiments assessments. The IoT-to-Edge-to-Cloud continuum of the IoSGT testbed extends from edge datacenter facilities running at the Federal University of Pará (UFPA), and the Federal University of Rio Grande do Norte (UFRN) premises, interconnected by RNP. An IoT-based off-the-shelf Smart Meter has been built by our group to provide a low-cost IoSGT Advanced Metering Infrastructure (AMI) use case. In addition to presenting the IoSGT architecture, our work includes prototyping and exercise on the testbed.