The following paper presents a novel orthogonal pilot design dedicated for \textcolor{black}{integrated sensing and communications (ISAC)} systems performing multi-user communications and target detection. After careful characterization of both sensing and communication metrics based on mutual information (MI), we propose a multi-objective optimization problem (MOOP) tailored for pilot design, dedicated for simultaneously maximizing both sensing and communication MIs. Moreover, the MOOP is further simplified to a single-objective optimization problem, which characterizes trade-offs between sensing and communication performances. Due to the non-convex nature of the optimization problem, we propose to solve it via the projected gradient descent method on the Stiefel manifold. Closed-form gradient expressions are derived, which enable execution of the projected gradient descent algorithm. Furthermore, we prove convergence to a fixed orthogonal pilot matrix. Finally, we demonstrate the capabilities and superiority of the proposed pilot design, and corroborate relevant trade-offs between sensing MI and communication MI. In particular, significant signal-to-noise ratio (SNR) gains for communication are reported, while re-using the same pilots for target detection with significant gains in terms of probability of detection for fixed false-alarm probability. Other interesting findings are reported through simulations, such as an \textit{information overlap} phenomenon, whereby the fruitful ISAC integration can be fully exploited.