Can you know in advance the path a message will take between you and another computer on the internet

No, you really cannot tell.

Explanation:

Packets on a network are always taking different routes or paths. Based on the traffic, the router will choose the best path to send packets to. For instance, if the network experiences redundancies or network congestion between the host and destination devices, additional paths will always be available. Unless we have manually configured proxies, we will never tell in advance the path a message will take. Internet routes are dynamic in nature especially since they are based upon traffic loads among other factors