This paper shows how data-driven deep generative models can be utilized to solve challenging phase retrieval problems, in which one wants to reconstruct a signal from only few intensity measurements. Classical iterative algorithms are known to work well if initialized close to the optimum but otherwise suffer from non-convexity and often get stuck in local minima. We therefore propose DeepInit Phase Retrieval, which uses regularized gradient descent under a deep generative data prior to compute a trained initialization for a fast classical algorithm (e.g. the randomized Kaczmarz method). We empirically show that our hybrid approach is able to deliver very high reconstruction results at low sampling rates even when there is significant generator model error. Conceptually, learned initializations may therefore help to overcome the non-convexity of the problem by starting classical descent steps closer to the global optimum. Also, our idea demonstrates superior runtime performance over conventional gradient-based reconstruction methods. We evaluate our method for generic measurements and show empirically that it is also applicable to diffraction-type measurement models which are found in terahertz single-pixel phase retrieval.