i am anglo-argentine so get the best of both worlds, in that i can hang out with british or american friends and be as foreign as the next person, then the following day be taken to cool and off the beaten track places and events by my argie friends and be
una porteña más as i speak fluent argie spanish and don't look like a tourist.
two notable recommendations to get a feel of a traditional argentine night is to do the 'bar de roberto'
http://www.barderoberto.com.ar/and when that closes walk the short distance half a block away to 'el conventillo de teodoro'
http://11870.com/pro/conventillo-teodoro/map. both highly recommended, both serving drink and the latter empanadas and finger food, where old cronies get up and do impromptu jams of milongas well into the morning hours. i've left there at 8am before and things were still rolling.
this is abasto incidentally, so you don't have to go beyond the panamericana for something fun, cheap and very much off the tourist circuit. both have got a nice mixed crowd which is an added bonus, lots of 20 somethings but the porteño old timers frequent these places too.