There are 4 riding clubs in Buenos Aires - Club Hipico (on Alcorta, next to River stadium), Club Aleman and Buenos Aires and GEBA (which I think is affiliated with Aleman now). Those 3 are on Dorrego, right by Lugones (behind the racetrack). All of them have a riding school.
I believe after a short amount of time (might be 4 lessons?), you have to become a member of the club and pay the monthly membership fee.
Hipico is the most expensive and has the nicest amenties (restaurant, pool, gym, etc). Very prestigious and they have a good riding school although the director just retired so I'm not sure who is taking over. Most of the top horse shows are held at Hipico. Aleman has the best footing and a good program with lots of students. They also have a program and do lots of schooling shows and its slightly less expensive. Buenos Aires is okay although it's my least favorite of them. GEBA used to be the most low-key and was the cheapest (no real ring there, very limited facilities) but as I mentioned, I think it's now affiliated with Aleman so you have to pay the Aleman membership fee but you can use their facilities and rings.
Just an FYI, if your daughter is interested in showing, all shows are canceled because of the outbreak of EVA. I don't think shows will be back on until August. They probably will have schooling (internal) shows.