First things first: I was in Argentina for two months. They are a culture heavily influenced by the Italians and Germans. Therefore, they are very much a bread, pasta, pastry & pizza culture. Eating here with a gluten & lactose allergy was challenging. A typical "porteno" breakfast is a medialuna (pastry/croissant) with jam and some tea. Obviously this was a problem. So, I ate loads of fruit for breakfast - apples, bananas, oranges. While I realize I was seriously lacking in protein in the morning, this was for a fairly short period of time, and I just dealt with it.
Lunch - ok so sandwiches are huge in Argentina. Again, this is a problem. While gluten-free isn't new here, there really aren't too many options. My typical lunch was anything from rice to arabian food to a quarter of a chicken to mongolian BBQ. So, there are choices, but you have to be aware of what ingredients go into the food your eating. This is where your pro-active research needs to come into play.
Dinner - Fortunately, the people I stayed with in Buenos Aires were amazing and worked around all my annoying allergies. Vegetable soup was a staple in our house. STEAK! The steak is world-renown in Argentina and for good reason. I NEVER got one that was over or under cooked. It was always perfectly medium rare. Yum.
One last note: BE WARY OF STREET FOOD! Street food is amazing!! MOST OF THE TIME! Try everything and have fun, just be very careful and probably carry some pro-biotics with you.