Buenos Aires is known as the "Paris of the South". It has beautiful allees, great shopping and of course the Tango in the colorfull city part "La Boca"! Note that Buenos Aires is different from the rest of Argentina.
The mighty Iguazu falls in the jungle are one of the natural world wonders. They are definitely worth visiting if you are in the area (less than 1000kms away!). You need to spend at least 1 day on the Brasilean and 1 day on the Argentinean side.
The Aconcagua at 6,961 metres, in the Andes mountains between Santiago de Chile and Mendoza, is the highest mountain outside the Himalayas.
Bariloche is known as "little Switzerland". Heidi does not live here nor is there a Matterhorn, but there is lots of Chocolate and great scenery. The lakes could indeed be Lucerne, but less crowded...
Cafayate is an important tourist location in Northern Argentina thanks to the extraordinary landscape and the great wines. The largest golf course in S-America is here.
Mendoza is a nice, modern city at the foot of the Andes. It has many vineyards and bodegas. Feel free to taste and enjoy the sites. From Mendoza you can also cross to Chile passing the Aconcagua mountain.
Puerto Madryn is a coastal city in Patagonia. It is well-known for the nearby Valdes Peninsula with a lot of "wild" life, if you can call all those sweet Penguins and chubby Sea Elephants "wild"... You may have a real opportunity to see Orcas here.
A personal favorite. Nice "little" town with flowers and churches in an arid environment "away from it all". Salta's historic center is a UNESCO World Heritage Site and one of the most beautiful in all of Argentina.