<p>Since I'm not a huge fan of any of the current solutions for managing the resources and knowledge that I have, I was thinking about making my own solution, which will involve custom code as well as possible integration of FOSS solutions. I would start development on my local machine, but if I like it, how difficult would it be to migrate over to a public server and let others also use this tool? What kinds of challenges might I be facing?</p>

In theory, nothing, beyond just the process of moving stuff to the new machine. You can set up your own servers, on your own ports (port 80 for example).
You can even create your own fake domain at home, with just a tweak to the /etc/hosts files (or the equivalent on Windows).
Now, if you're developing on Windows and hosting on unix, you'll have platform issues, so I'd suggest against that, at least for a first project.
But other than that, it should be straightforward.