How to "publish" without dreamweaver

Context:

I was recently put in charge of the website for my organization. The current site is only static HTML and we need some database backed stuff. I can handcode HTML, CSS, and Javascript. I understand how to make a server and all of that but the current website was managed via dreamweaver, which I have never used. The site is hosted on a server I don't control (belongs to our parent organization who are not set up to handle this kind of thing on their server). I want to just throw a redirect on the old site and manage my own server or get hosting that allows me to control the back end. My predecessor told me to hit publish on dreamweaver and fill in the SFTP Address, Username, Password, Root Directory, and Web URL. I get what is going on here but I have no clue how to do this without dreamweaver. Their must be a way. I could get the trial of dreamweaver and use it to make this one change but it makes me uncomfortable that after the month I will be unable to access a page I am responsible for. This feels like something very basic I should know so I am sorry if this is a ridiculous question. Thanks in advance.

Cut to the question:

How do I replicate the "publish" action of dreamweaver if I have the SFTP Address, Username, Password, Root Directory, and Web URL of the target server?