Caching at the network edge has emerged as a viable solution for alleviating the severe capacity crunch in modern content centric wireless networks by leveraging network load-balancing in the form of localized content storage and delivery. In this work, we consider a cache-aided network where the cache storage phase is assisted by a central server and users can demand multiple files at each transmission interval. To service these demands, we consider two delivery models - $(1)$ centralized content delivery where user demands at each transmission interval are serviced by the central server via multicast transmissions; and $(2)$ device-to-device (D2D) assisted distributed delivery where users multicast to each other in order to service file demands. For such cache-aided networks, we present new results on the fundamental cache storage vs. transmission rate tradeoff. Specifically, we develop a new technique for characterizing information theoretic lower bounds on the storage-rate tradeoff and show that the new lower bounds are strictly tighter than cut-set bounds from literature. Furthermore, using the new lower bounds, we establish the optimal storage-rate tradeoff to within a constant multiplicative gap. We show that, for multiple demands per user, achievable schemes based on repetition of schemes for single demands are order-optimal under both delivery models.