I know that it probably isn't, but is it possible to write/read a file on the server without server side languages.
At a fairly abstract level: You have to have a server. That server has to be written in some language.
All regular web servers support the reading of files from file systems. Generally you specify a directory to be the root, then the local part of the URL gets mapped on to that directory and its sub-directories.
Writing is trickier. The HTTP specification includes the PUT verb, but not many servers have built in support for doing anything useful with it … and you'd almost always want to add some sort of access control before allowing anyone with access to the Internet to write files.
Expressing the logic behing the access control and where the file should be placed is something that is usually most easily handled with a programming language.
I've looked in a bunch of places and found oblique references to reading files on the server using XmlHttpRequest. I unfortunately have not found any "good" info on reading files.
XMLHttpRequest is just the standard API that browsers provide to JavaScript for making HTTP requests. It is entirely client side. You give it a URL, you get the data the server returns in a callback.
I have found absolutely nothing when it comes to writing files without server side languages though. I am somewhat new to coding (4 to 5 months) and started out with HTML/JS, so I don't know any server side languages and when I look at them I get a headache trying to understand.
JavaScript is a server side language. It isn't the easiest one to get to grips with, but it is an option. See node.js.
I'd recommend getting to grips with Python and using Django for someone new to programming.
I lean towards Perl and Catalyst for my own projects.
I know AJAX interacts with the server, so I thought there must be some way to write a file on the server with AJAX.
Ajax just means "Making an HTTP request from JavaScript without leaving the page". You can send a file, but once the request gets to the server, the server has to handle it.