Global transport and communication networks enable information, ideas and infectious diseases now to spread at speeds far beyond what has historically been possible. To effectively monitor, design, or intervene in such epidemic-like processes, there is a need to predict the speed of a particular contagion in a particular network, and to distinguish between nodes that are more likely to become infected sooner or later during an outbreak. Here, we study these quantities using a message-passing approach to derive simple and effective predictions which are validated against epidemic simulations on a variety of real-world networks with good agreement. In addition to individualized predictions for different nodes, we find an overall sudden transition from low density to almost full network saturation as the contagion develops in time. Our theory is developed and explained in the setting of simple contagions on tree-like networks, but we are also able to show how the method extends remarkably well to complex contagions and highly clustered networks.