Palermo is a big neighbourhood.
There is a big, outdoors, clean swimming pool, open 7 days a week, which is also affordable:
nstituto de Natacion y Deportes
Avda. Juan B. Justo 1423
Palermo - Capital Federal - Ciudad de Buenos Aires
Telefono: (54 11) 4771-7873 4777-1472
Categorias: Educacion Superior (Terciaria) No Universitaria Privada
Website:
http://www.natacionydeportes.com.ar
Their purpose is to teach to swim teachers, but you can go and swim by yourself.
Disco is a good place to go for weekly supplies, but has the brandname to be the most expensive supermarket, only Jumbo being a little more expensive. There are many Coto and Carrefour supermarkets in Palermo, with variety and better prices. What about the ones run by chinese people, also knowns here as chino?
Keep in mind that the main supermarkets like Disco, Coto and Carrefour have arrangements with local banks and its credit cards, so if you go one day and you have to way long lines to pay, it is because there is a discount going on. Also Disco gives some discount to the jubilados on Wednesdays. These 3 supermarkets deliver the merchandise to your house, but their schedule is very flexible.
For bigger quantities of food, you might try to go to Mercado Central, which is on the way to Ezeiza airport. But you need to own a car or arrange with someone to share the expenses of a car. I read an article that there are buses charter style connecting Palermo to Mercado Central, but not sure about it.