First-order Policy Gradient (FoPG) algorithms such as Backpropagation through Time and Analytical Policy Gradients leverage local simulation physics to accelerate policy search, significantly improving sample efficiency in robot control compared to standard model-free reinforcement learning. However, FoPG algorithms can exhibit poor learning dynamics in contact-rich tasks like locomotion. Previous approaches address this issue by alleviating contact dynamics via algorithmic or simulation innovations. In contrast, we propose guiding the policy search by learning a residual over a simple baseline policy. For quadruped locomotion, we find that the role of residual policy learning in FoPG-based training (FoPG RPL) is primarily to improve asymptotic rewards, compared to improving sample efficiency for model-free RL. Additionally, we provide insights on applying FoPG's to pixel-based local navigation, training a point-mass robot to convergence within seconds. Finally, we showcase the versatility of FoPG RPL by using it to train locomotion and perceptive navigation end-to-end on a quadruped in minutes.