Near El Calafate and El Chalten, the Los Glaciares National Park is a world-renowned climbing destination in Patagonia. It also has world class hiking routes. The best routes include Laguna Torre (12 mile out & back with 650 feet of vertical), Loma del Pliegue Tumbado (12 mile out & back with 3,400 feet of vertical), Laguna Capri (8 mile loop with 1,000 feet of vertical), Lago Electrico (9 mile out & back), Laguna de Los Tres to El Pilar (12 miles with 2,300 feet of vertical), and the popular Icefield Circuit (47 mile loop with 5,000 feet of elevation change).
For exploring Los Glaciares National Park, the newer, circa 1986, town of El Chalten makes a great base.
Keep in mind that this region of Patagonia is known for it's high winds and rapidly varying weather (NOAA does provide a good forecast for the region). Always carry a Gortex waterproof layer. While there are some English-speakers in El Chalten, a basic understanding of Spanish is essential.
To reach Los Glaciares National Park from Buenos Aries, fly from Aeroparque Airport to El Calafate. Once in El Calafate, there is a bus from El Calafate to El Chalten (reservation required).