Spins associated to solid-state color centers are a promising platform for investigating quantum computation and quantum networks. Recent experiments have demonstrated multiqubit quantum processors, optical interconnects, and basic quantum error-correction protocols. One of the key open challenges towards larger-scale systems is to realize high-fidelity universal quantum gates. In this work, we design and demonstrate a complete high-fidelity gate set for the two-qubit system formed by the electron and nuclear spin of a nitrogen-vacancy center in diamond. We use gate set tomography (GST) to systematically optimize the gates and demonstrate single-qubit gate fidelities of up to 99.999⁢(1)% and a two-qubit gate fidelity of 99.93⁢(5)%. Our gates are designed to decouple unwanted interactions and can be extended to other electron-nuclear spin systems. The high fidelities demonstrated provide opportunities towards larger-scale quantum processing with color-center qubits.