There are tutorials out there in the vein of "Security for developers" or "Networking for developers" but how about DevOps for developers? Is there a "bible" of DevOps? A canonical introduction?
Basically, every time you run into something that should be someone else's job, do it.
You should definitely have your own servers to play around with. I use digital ocean which is pretty inexpensive but has an API to your account (example: https://www.digitalocean.com/community/articles/how-to-insta...).
I would look into using vagrant to spin up virtual machines associated with a particular software project. I find that helped me quite a bit with thinking of the server as being coupled to the application server software.
The reason that it's called "devops" instead of just "ops" is to create a separation between the type of ops where you don't EVER break anything and you move real slow and no one ever got fired for buying IBM, and the type of ops where everything's virtualized and software-defined, and things can, will, and should break without any penalty to the operation of the system as a whole.