Millimeter-wave MIMO systems have gained increasing traction towards the goal of meeting the high data-rate requirements in next-generation wireless systems. The focus of this work is on low-complexity beamforming approaches for initial UE discovery in such systems. Towards this goal, we first note the structure of the optimal beamformer with per-antenna gain and phase control and the structure of good beamformers with per-antenna phase-only control. Learning these beamforming structures in mmW systems is fraught with considerable complexities such as the need for a non-broadcast system design, the sensitivity of the beamformer approximants to small path length changes, etc. To overcome these issues, we establish a physical interpretation between these beamformer structures and the angles of departure/arrival of the dominant path(s). This physical interpretation provides a theoretical underpinning to the emerging interest on directional beamforming approaches that are less sensitive to small path length changes. While classical approaches for direction learning such as MUSIC have been well-understood, they suffer from many practical difficulties in a mmW context such as a non-broadcast system design and high computational complexity. A simpler broadcast solution for mmW systems is the adaptation of directional codebooks for beamforming at the two ends. We establish fundamental limits for the best beam broadening codebooks and propose a construction motivated by a virtual subarray architecture that is within a couple of dB of the best tradeoff curve at all useful beam broadening factors. We finally provide the received SNR loss-UE discovery latency tradeoff with the proposed constructions. Our results show that users with a reasonable link margin can be quickly discovered by the proposed design with a smooth roll-off in performance as the link margin deteriorates.