Short term rentals are from about $800 per month up:
http://www.bytargentina.com. I pay about $450 but that's because I am on a long term contract (min term of 2 years). Inflation is about 25% and its effects are noticable. Groceries/clothes/dinner/drinks cost either the same, or more than in the UK - the only things cheaper are property prices, rent, public transport. Cars and electrical items cost twice as much as in the UK.
Whilst Rio is more picturesque, surrounded by beaches, I found it a lot quieter during the week and a bit of a let down (more expensive than here too). Whereas in Buenos Aires you can go out any day of the week. And if the language is an issue:
http://www.pubcrawlba.com/ usually has big groups of both locals and English speaking.
http://www.fcbafa.com/en/ for regular football.
It's a great place to base yourself and go on mini trips around the continent from. Uruguay is very, very quiet, except punta del este during summer (dec-feb down here).
A lot of people just come to Buenos Aires and then move on to the next country. But there's so much variance here alone and places worth seeing as people have already said, particularly for the natural world (frozen South, desert/tropical North):
- El Calafate/Chalten (Perito moreno glacier and Fitz Roy, image search these places)
- Bariloche / Villa La Angostura (recovering from the Chilean volcano)
- Ushuaia
- Cordoba
- Mendoza
- Iguazu Falls
- Salta/Tucuman/Jujuy