Playing soccer for example is pretty cheap. Cost you about 10 to 15 peso a game and it's a good way to meet people and it takes you about an evening if you count with the traveltime, coming a bit earlier to talk with the lads and staying a bit after the game with a beer.
It's pretty easy to join up with an expat group on facebook. A local group is a lot more dificult
Joining Club Europeo is not that expensive(or maybe an American club if you are from the US) and it's good way to meet up with people.
Buy or bring rollerblades and go blading. There is also a website where you can meet up with other guys and girls who have this hobby and they also do trips from Palermo to Puerto Madero
Do some voluntary work
All not that expensive at it's good way to get to know people