On my trip to South America last year I spent a month there. On Sundays in the park early they have boot sales in Parque Centenario & Parque Rivadavia. Those might be a little out of your way as they took me about half an hour to get to from Palermo. They have the heat though at the best prices.
The Subways in the city are fast and clean and safe. Definitely hop on one and head to Callao y Corrientes.
That should lead you to a really good CD store called ZIVALS. No Records there, but if you travel down that street in the direction of Pizza Guerrin (Uruguay y Corrientes). There's a bunch of bookstores that have records, Also.. I can't remember the name of the street I think it is Rodriguez Pena (forgive my memory) but in walking distance theres like 3 nice record stores in there. If you get lost you're at one of the best Pizza places in the City - Pizzaria Guerrin.(Av. Corrientes 1368 (between Uruguay & Talcahuano). (I'm missing that Pizza as I think about it right now.)
There's one store in particular there in that neighborhood ran by a girl named Lucy that was the best spot I went to. I can't remember the name of. I have the business card here somewhere. I believe it was on Rodriguez Pena. They had a ton of soul records, and really great rock records. There was about 4 or 5 record stores total in that neighborhood, but her store was the best one. She had pictures of Hendrix everywhere. I have pictures, but my old hard drive kinda took a shit.
Don't waste your time looking for records underground around the Obelisk.. I wasted maybe 2 days trying to come up and got lost every single time. However my friend Mateo told me there is another good record store in San Telmo near Boca. Also.. don't waste your time at the Record stores in Palermo. They're mostly hipster spots that are overpriced.
Mateo is the drummer from the band Morbo y Mambo. He's a beat head who speaks english he might be able to give you more advice. I can send you his Facebook. You should also check out his band. Good shit!
Here's one of those record spot sites for Buenos Aires. (who knows how up to date it is). Where to buy records from Buenos Aires.