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.
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.
The Perito Moreno near Calafate is a gigantic glacier in the far south of Argentina. Unlike most glaciers this one is still growing. When big chunks of ice break of the head of the glacier into the lake, it is a great sound, something you won´t forget.
Ushuaia, an exotic name for a remote little city. You are at Tierra del Fuego, the most southern part of South-America. You may still see a few animals, but mostly it is wild, windy and abandoned. The Panamerican Highway that starts in Alaska ends here... Boat expeditions to the Antarctic may depart here.
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.