The properties of Kitaev materials are attracting ever increasing attention owing to their exotic properties. In realistic two-dimensional materials, the Kitaev interaction is often accompanied by the Dzyaloshinskii-Moriya interaction, which poses a challenge for distinguishing their magnitudes separately. In this paper, we demonstrate that it can be done by accessing magnonic transport properties. By studying honeycomb ferromagnets exhibiting Dzyaloshinskii-Moriya and Kitaev interactions simultaneously, we reveal nontrivial magnonic topological properties accompanied by intricate magnonic transport characteristics as given by thermal Hall and magnon Nernst effects. We also investigate the effect of a magnetic field, showing that it does not only break the symmetry of the system but also brings drastic modifications to magnonic topological transport properties, which serve as hallmarks of the relative strength of anisotropic exchange interactions. Based on our findings, we suggest strategies to estimate the importance of Kitaev interactions in real materials.