For a UX newbie you already have the right mindset. Don't do a complete makeover when the website or application is not working as well as expected, but try changing small things first.
From your question I get the feeling there is even some anxiety for performing any usability tests and such. Sure this is fueled by inexperience, but it might also be fueled by lack of equipment or settings.
Good thing is you don't need expensive equipment like eye-tracking stuff. Usability tests can be done by simply observing people using the application.
These people can be your friends or even you! You are a student at that school. You have used that program to enroll in class. You know where, in your opinion, the problems lie.
Your friends are users too. You might think they're not fit test persons, because they are like you. But they too are students at that school (I presume) and they too are real users.
If you want to add some dynamic to the test persons by adding someone who is less tech savvy, there is no shame in asking someone in the family with less computer skills to test it.
If the situation allows you, you can even ask a person you just saw performing the enrollment task what their experience was, where they got stuck or what they think was unclear. Asking them what could be improved is dangerous. They often don't really know. You're supposed to be the expert that can think of a solution to their problems.
But how do you test your improvements?
You mention you can't implement the changes in order to test it. The thing is however, you don't need the real thing to test improvements. It would be most ideal if you could use the real data and test the improvements while the old version is still in use (try reading into A/B testing; it might be a solution for you). But just below that ideal situation is a demo where you simulate the enrollment flow. Not every single link has to work and not everything has to be real data, just as long as the steps the users take are the same.
And below that are more static prototypes. These could even be images of the new situation. Let your test persons point at the part where they think they should click next and then show them the screen behind that button by presenting them with a new image.
It is easier to use any of many prototype tools online that let you link images by laying invisible hitzones over the buttons and links and linking that to the corresponding image.
Compare the time the users need to perform the task in the original application with the time they needed to complete the task in your improved version and compare the amount of clicks.
Even after all that you can't expect to present the school (or the person responsible for the application) with a perfect solution to all their troubles. You can't always substantiate decisions with actual numbers and facts. Often you don't know your solution will be the right solution.
But how do you convince (let's name it) the client?
You do that by showing them what is wrong with their application in the first place. Try to be gentle about it, because they might be the one to have build it in the first place. But show them user opinions and the test results of the original application. Then show them your solution and tell them how it would solve those problems and show them the better time scores.
