I’m starting my 21st year in education technology with this upcoming year. Moving up to more responsibilities takes patience and keeping your eye on the long-term, and not comparing yourself to others who work in private sector. Working in schools, you won’t advance as quickly or as high as others who are in the corporate or small/medium business world, but the rate of job-churn is much less volatile.
I was one of the few among my friends who weathered the economic crash of around 2006-2009.
Honestly, it does take self-study and self-discipline to move up if you’re in the education world, even to amass enough experience to get out of education. That study, however, doesn’t have to involve formal training or classroom experience for yourself. There are plenty of free resources out in the world.
The first issue, though, is to identify what you want to do and to focus your direction. “I want to make more money” and “I want to move up in my career” are both too vague and also not specific to IT. “I want to be a server administrator with a focus on Windows Server and Linux” is much more specific and you’ll get more useful information that way.
Second, prioritize your studies. At 20 years old, it’ll be more tempting to check out at the end of the day and go hang out with friends until 1:00am or be online playing games until 3:00am (I’m generalizing here, obviously). If you’re passionate about learning WinServer 2016 or even 2019, you’ll make the time for it. But you have to make that decision; it’s not going to be assigned to you.
Third, look at the job postings that interest you and see what skills you’re lacking if you were to apply for those jobs (I highlight this to distinguish it from “a job that pays more”). Ignore the “3-5 years experience in …” because you’re not going to get those jobs anyway. Look at the technical skills they list off and focus on those.
Fourth, after you’ve built your study plan, start incorporating a few of those things into your job. Ask for permission first from your supervisor, obviously. Approach it from the angle of “I want to help the school,” NOT “I want to help myself.” I often see too much of the latter in these requests for career-advancement advice topics 
If you’re told “no,” accept it with grace and try again later with something else.
If you have a good relationship with your supervisor, ask for advice on how you can help with more than just “open ticket, reboot computer, close ticket.” Your boss may see some deficiencies in your skills (or even attitude) that you’re not aware of. Take that as constructive criticism and as an opportunity to improve.
Finally, this won’t be what you want to hear, but expect to give yourself 5 years to build up your experience and skills before other entry-level positions start opening up for you. In my 20 years in schools, I’ve watched and guided a few younger techs grow and move on, but it took a few years, some cold doses of reality, and a lot of self-study on their part to find their niche.
We had a kid we’d kept an eye on since he was in middle school/junior high become the lead network administrator for a major transportation company soon after graduating from high school. But we had to mold him into something more acceptable for the adult working world, because even though he had the technical skills and ambition, he lacked self-control and was very arrogant toward regular users who weren’t in IT. That just came from his lack of experience with people outside of school, though.
We had another kid we brought on board starting as a high school intern. He also grew up and got metaphorically smacked by the adult world and has since matured into a respected admin and leader. After 12-13 years of working in the department, he’s now the supervisor for all of the field technicians who came in after him.
Going further back in time, soon after I was hired in my previous school, we hired a tech fresh out of his own high school internship. He took it upon himself to spend years learning PowerShell, system deployment/management, and so on. 14 years after his hire, he moved on to a bigger school organization, and is now one of the leaders in the state for PowerShell and technology management in public education.
18 years after I started in education technology, I’m now in data analytics and business intelligence reporting. But I had to build that foundation myself through self-study and after-hours projects on my own.
So the point of this long-winded narrative is that it’s certainly possible to grow out of where you are. But it takes a lot of initiative on your own plus more patience and an eye toward the long-term goal than you might expect. There’s also more than just technical skills involved. Finding a mentor (even if it’s your boss) will be more beneficial than you might imagine.
