Training recurrent neural networks (RNNs) on low-dimensional tasks has been widely used to model functional biological networks. However, the solutions found by learning and the effect of initial connectivity are not well understood. Here, we examine RNNs trained using gradient descent on different tasks inspired by the neuroscience literature. We find that the changes in recurrent connectivity can be described by low-rank matrices. This observation holds even in the presence of random initial connectivity, although this initial connectivity has full rank and significantly accelerates training. To understand the origin of these observations, we turn to an analytically tractable setting: training a linear RNN on a simpler task. We show how the low-dimensional task structure leads to low-rank changes to connectivity, and how random initial connectivity facilitates learning. Altogether, our study opens a new perspective to understand learning in RNNs in light of low-rank connectivity changes and the synergistic role of random initialization.