It doesnt hurt to have the paperwork in order.
It is most likely that you are not going to need a CUIL to work as en English teacher.
You do not need any kind of visa, as you are a citizen of this country. You need patience, be aware that when you go to a public office the employees will drink mate and galletitas for one hour before starting to work with the public, probably they will tell you to visit the web page to get information (and the information is not available on line), and to make it easy, you are going to deal with bouracrats. Good luck!
