We propose a simple scheme to generate quantum entanglement and one-way steering between distinct mode pairs in a generic cavity magnomechanical system, which is composed of a microwave cavity and a yttrium iron garnet sphere supporting magnon and phonon modes. The microwave cavity is pumped by a weak squeezed vacuum field, which plays an important role for establishing quantum entanglement and steering. It is found that when the magnon mode is driven by the red-detuned laser, the maximum entanglement between cavity mode and phonon mode and the maximum phonon-to-photon one-way steering can be effectively generated via adjusting the ratio of two coupling rates. While under the much weaker magnomechanical coupling, the quantum entanglement and one-way steering between cavity mode and magnon mode can be achieved, where the steering direction is determined merely by the relative dissipation strength of the cavity to the magnon mode. More interestingly, we reveal that the robustness to the temperature for entanglement and steering between any mode pairs can be evidently enhanced by selecting the squeezing parameter appropriately.