I'm trying to get a feel for how complex Tableau Server is to install and become familiar with. If someone is comfortable managing AWS is it quickly within reach from available documentation and community support?

It takes time, but yes, you can definitely learn it, assuming you are also comfortable with the OS you're going to be using. The install is relatively straightforward and the documentation is good. I'd definitely recommend looking into some training and, potentially, certification, as that will ensure you have the base knowledge. If you are going to immediately be running a production server, then you may want to look into getting some help until you're ready to own it yourself.