Plan when you want to go. They recommend to go in the summer (December to February) when it's dry. We went in early November when the crowds have not arrived yet, and except for a light drizzle on a couple evenings, the sun was out so we were able to see the areas major stunners - Perito Moreno Glacier and Fitz Roy - in a backdrop of sunny blue skies. The one drawback though (if you can call it that) we were too early to see Laguna de los Tres - it was still very much frozen and we were unable to see the famed aquamarine waters framing Fitz Roy.
There are plenty of direct flights leaving from Buenos Aires in Argentina from airlines like Aerolinas Argentinas and LATAM. Be prepared for flight delays though as they are not uncommon.
Most people (us included!) don't have much time to enjoy this wild place. We just had enough time to see the highlights of Los Glaciares National Park: 1. Perito Moreno Glacier, and 2. Laguna de Lost Tres right below Fitz Roy. If we had more time, we would definitely do the hike to Laguna Torre.
The gateway town to Perito Moreno Glacier is El Calafate, where the airport is. There are a lot of accommodation options here, from hostels to some fancier hotels. We stayed at Mirador del Lago and found it to be plenty spacious and comfortable. Our room included daily breakfast and partial views of Lago Argentina. It was a bit away from the central part of downtown, but we didn't mind the walk. Being an well-trafficked mountain base town, there were a lot of restaurants and shops, as well as tour companies and ATMs. We didn't run into any problems using our credit cards here.
For more about El Calafate, see related post here.