We consider an infinite system of particles on the integer lattice Z that: (1)migrate to the right with a random delay, (2) branch along the way according to a random law depending on their position (random medium). In Part I, the first part of a two-part presentation, the initial configuration has one particle at each site. The long-time limit exponential growth rate of the expected number of particles at site 0 (local particle density) does not depend on the realization of the random medium, but only on the law. It is computed in the form of a variational formula that can be solved explicitly. The result reveals two phase transitions associated with localization vs. delocalization and survival vs. extinction. In earlier work the exponential growth rate of the Cesaro limit of the number of particles per site (global particle density) was studied and a different variational formula was found, but with similar structure, solution, and phases. Combination of the two results reveals an intermediate phase where the population globally survives but locally becomes extinct (i.e., dies out on any fixed finite set of sites).