Quantum tomography is an essential method of the photonic technology toolbox and is routinely used for evaluation of experimentally prepared states of light and characterization of devices transforming such states. The tomography procedure consists of many different sequentially performed measurements. We present considerable tomography speedup by optimally arranging the individual constituent measurements, which is equivalent to solving an instance of the traveling salesman problem. As an example, we obtain solutions for photonic systems of up to five qubits, and conclude that already for systems of three or more qubits, the total duration of the tomography procedure can be halved. The reported speedup has been verified experimentally for quantum state tomography and also for full quantum process characterization up to six qubits, without resorting to any complexity reduction or simplification of the system of interest. Our approach is versatile and reduces the time of an input-output characterization of optical devices and various scattering processes as well.