I would like to monitor the data that gets sent to and from an application running on my computer, to this outside server.
What do you suppose is a suitable application to best perform this process??
In addition to using Wireshark, you could also perform hooking on socket functions but I am afraid that I can't provide you with any suitable reference of piece of software that let you do so.
Therefore, I suggest you to debug the application, setting breakpoints toward the calls to the interesting functions (I mean send(), recv() and the like) or write a DLL - unless you're not running on Windows - that will hook the functions once you've injected it into your running process.