Algorithms for effectively routing messages from a source to multiple destination nodes in a store-and-forward computer network are studied. The focus is on minimizing the network cost (NC), which is the sum of weights of the links in the routing path. Several heuristic algorithms are studied for finding the NC minimum path (which is an NP-complete problem). Among them are variations of a minimum spanning tree (MST) heuristic and heuristics for the traveling salesman problem, both of which use global network information. Another set of heuristics examined are based on using only the shortest paths to the destinations. While the MST algorithm has the best worst case performance among all algorithms, a detailed, empirical study of the "average" performance of the algorithms on typical, randomly chosen networks reveals that simpler heuristics are almost as effective. The NC cost measure is also compared to the destination cost (DC), which is the sum of weights of the shortest path distances to all destinations. A scheme of algorithms is given which trades off between NC and DC.