We designed, prototyped, and experimentally demonstrated, for the first time to our knowledge, an optoelectronic spiking neuron inspired by the Izhikevich model incorporating both excitatory and inhibitory optical spiking inputs and producing optical spiking outputs accordingly. The optoelectronic neurons consist of three transistors acting as electrical spiking circuits, a vertical-cavity surface-emitting laser (VCSEL) for optical spiking outputs, and two photodetectors for excitatory and inhibitory optical spiking inputs. Additional inclusion of capacitors and resistors complete the Izhikevich-inspired optoelectronic neurons, which receive excitatory and inhibitory optical spikes as inputs from other optoelectronic neurons. We developed a detailed optoelectronic neuron model in Verilog-A and simulated the circuit-level operation of various cases with excitatory input and inhibitory input signals. The experimental results closely resemble the simulated results and demonstrate how the excitatory inputs trigger the optical spiking outputs while the inhibitory inputs suppress the outputs. Utilizing the simulated neuron model, we conducted simulations using fully connected (FC) and convolutional neural networks (CNN). The simulation results using MNIST handwritten digits recognition show 90% accuracy on unsupervised learning and 97% accuracy on a supervised modified FC neural network. We further designed a nanoscale optoelectronic neuron utilizing quantum impedance conversion where a 200 aJ/spike input can trigger the output from on-chip nanolasers with 10 fJ/spike. The nanoscale neuron can support a fanout of ~80 or overcome 19 dB excess optical loss while running at 10 GSpikes/second in the neural network, which corresponds to 100x throughput and 1000x energy-efficiency improvement compared to state-of-art electrical neuromorphic hardware such as Loihi and NeuroGrid.