Variability on the external conditions has important consequences for the dynamics and organization of biological systems, and, in many cases, the characteristic timescale of environmental changes as well as their correlations play a fundamental role in the way living systems adapt and respond to it. A proper mathematical approach to understand population dynamic, thus, requires of approaches more refined than e.g. simple white-noise approximations. To shed further light onto this problem, in this paper we propose a unifying framework based on different analytical and numerical tools available to deal with "colored" environmental noise. In particular, we employ a "unified colored noise approximation" to map the original problem into an effective one with white noise, and then we apply a standard path integral approach to gain analytical understanding. For the sake of specificity, we present our approach using as a guideline a variation of the contact process -which can also be seen as a birth-death process of the Malthus-Verhulst class-where the propagation/birth rate varies stochastically in time. Our approach allows us to tackle in a systematic manner some of the relevant questions concerning population dynamics under environmental variability such as, for instance, determining the stationary population density, establishing the conditions under which a population may become extinct, and estimating extinction times. More in general, we put the focus on the emerging phase diagram and its possible phase transitions, underlying how these are affected by the presence of environmental noise time-correlations.