What determines the flow rate through a pipe when exiting a state of rest?

The standard tool for this type of problem is Bernoulli's equation, which relates the velocity of the water as it enters the pipe to its velocity as it exits the pipe and the pressure difference at the two ends. In addition, if you assume that the water is incompressible (a pretty good approximation for water assuming you're not using stupidly large pressures), then the flow rate $Q$ into the pipe must equal the flow rate out of the pipe. This implies that $Q = Av$ is the same at both ends (where $A$ is the cross-sectional area of the pipe.) These equations can then be solved to find the flow rate.

All of this assumes, however, that your pipe is in steady state, i.e., there is a constant flow rate being maintained by a constant pressure head. If you are in a situation where the pressure head is not staying constant, then things get more complicated (and beyond my ability to dash off a quick answer.)