How do I push changes from my developer subdomain to my FQDN?

Long story short, I was put in charge of managing my robotics team's website this year, and I've realized that I'm in way over my head. I really haven't had any guidance or someone to train me on how our website works, they kind of just told me what my job was (update our website throughout the year) and left me to do it.

One piece of advice that I got was to test/preview my changes on my dev subdomain before making changes to my main FQDN, and I'm going to be honest I don't really know what that means. I've had some trouble the past couple days though where I accidentally put changes on my remote server that I shouldn't have.

I guess my question is how do I create some sort of version control system with my dev subdomain where I can test changes there and then push them to my main domain? Should I use an FTP client? I'm not sure. Could someone help me out? 

Thanks,

A struggling high school student in much need of help.