What should I do/know when I arrive at the airport?

Ezeiza is pretty far from Buenos Aires You can take an Uber, Taxi or some vans called "Tienda Leon" to the city

What's different from other countries/cities?

Buenos Aires has some dangerous neighbourhoods, specially for tourists. Please don't wander alone and make sure where you are going is safe.
Most stores open up at 10am.

How can I rent an apartment?

If you are just going to stay for a couple of days then Airbnb dhould be fine. Else, for longer stays you can look into zonaprop.com.ar for apartments.

I need a bank account / debit card

The banks with the best digital experience and most number of branches are Banco Galicia and Banco Santander. Just visit one of their branches and they'll hook you up. Beware, banks are only opened from 10am to 3pm in Argentina.

Where can I workout?

Search for Megatlon or Sportclub on Google Maps. They have many branches and are the best equipped gyms here.

Where can I go for a weekend getaway?

Mar del Plata, Pinamar, Carilo if you like beaches.
Colonia del Sacramento (uruguay) for something romantic or if you like historic places.