We consider level-set percolation for the Gaussian membrane model on $\mathbb{Z}^d$, with $d \geq 5$, and establish that as $h \in \mathbb{R}$ varies, a non-trivial percolation phase transition for the level-set above level $h$ occurs at some finite critical level $h_\ast$, which we show to be positive in high dimensions. Along $h_\ast$, two further natural critical levels $h_{\ast\ast}$ and $\overline{h}$ are introduced, and we establish that $-\infty <\overline{h} \leq h_\ast \leq h_{\ast\ast} < \infty$, in all dimensions. For $h > h_{\ast\ast}$, we find that the connectivity function of the level-set above $h$ admits stretched exponential decay, whereas for $h < \overline{h}$, chemical distances in the (unique) infinite cluster of the level-set are shown to be comparable to the Euclidean distance, by verifying conditions identified by Drewitz, Ráth and Sapozhnikov, see arXiv:1212.2885, for general correlated percolation models. As a pivotal tool to study its level-set, we prove novel decoupling inequalities for the membrane model.