The first thing to do before starting the long process of residency in Argentina is to do some preliminary background research. This involves talking to other expats that hold the same passport(s) that you do, searching google, and checking out the expat forums (I like www.baexpats.com), and of course -- use this blog!
It is important to think about how long you think you will really be in Argentina. The process is long and frustrating so you need to be committed to idea of staying here to really make it worth it.
For example, if you plan to be here short-term to teach English maybe just getting the CUIT as a monotributista is an OK solution and it would not be necessary to seek residency. You can be a tourist in Argentina for a LONG time before anyone starts asking questions and trip to migraciones every 3 months and Uruguay every 6 months is not that difficult.
However, there are many advantages to a CUIL and DNI. For me, I want the ability to have a "real job" and be paid legally. Without papers you will likely have to work "en negro" which means you are paid "under the table", tax-free and in cash.
Alright now that you have decided you want residency let´s keep moving forward on how to initiate the process.