The increasing demand for high-volume multimedia services through mobile user equipment (UEs) has imposed a significant burden on mobile networks. To cope with this growth in demand, it is necessary to extend the 5G network's ability to meet quality-of-service (QoS) requirements. The integration of Multi-access Edge Computing (MEC) with 5G technology, 5G-MEC, emerges as a pivotal solution, offering ultra-low latency, ultra-high reliability, and continuous connectivity to support various latency-sensitive applications for UEs. Despite these advancements, the mobility of UEs introduces significant spatio-temporal uncertainties, posing a major challenge on optimizing content delivery routes and directly impacting both latency and service continuity for UEs. Addressing this challenge necessitates suitable approaches for selecting optimal 5G-MEC components, with the goal of minimizing latency and reducing the frequency of handovers, ultimately ensuring a seamless content delivery experience. This paper proposes two learning-based approaches to tackle the problem of 5G-MEC component selection to facilitate QoS-aware content delivery in the absence of complete information about the dynamics of the 5G-MEC environment. First, we design an online sequential decision-making approach, called QCS-MAB, to decide on the content delivery routes in real-time while achieving a bounded performance. We then propose a deep learning approach, called QCS-DNN, to efficiently solve large-scale 5G-MEC component selection problems. We evaluate the effectiveness of our proposed approaches through extensive experiments using a real-world dataset. The results demonstrate that both QCS-MAB and QCS-DNN achieve near-optimal latency and significantly reduced handover times, significantly enhancing the 5G-MEC content delivery experience.