I am not an expert, it would be nice to hear the answers of an expert, also because I am planning to return to Buenos Aires around September for a couple of months and I want to dance tango as well.
I only know of two places.
A very touristic place to dance tango is La Viruta (Armenia 1366):
http://www.lavirutatango.com
It is very informal and included in the price of the classes is the entrance to the milonga (i.e. when they dance tango, not the type of dance).
I think you can give la Viruta a try, because you can combine your first class with a milonga afterwards. Around 10 - 11 PM, the milonga is still full of inexperienced people of which many are also dancing tango for the first time. If you want to dance, I advise you to dance early in the evening. It could be nice to stay until 3 AM or later, just to watch, because the later in the evening, the more experienced the dancers. You probably will not be dancing at 3 AM anymore, because you'd be intimidated by how the other people are dancing.
Personally, I think La Viruta is a nice place to go to a milonga as a tourist, because it is very informal. However, I think the classes are not the best quality, because they teach a lot of people at the same time. At the beginning of the milonga, the teachers show off how they dance, which I think is not necessary, but maybe that is what tourists want to see or the teachers need to boost their ego, whatever.
You can check the schedule on the Internet, it seems they have classes every day except Monday and I guess the milonga is after the classes.
The other place I danced at is
http://www.mayoralyelsamaria.com.ar/
near Callao/Santa Fe. They have no milonga there, but classes were better than at la Viruta. Also the building is nice. There are group classes each Monday, Wednesday and Friday. I think it is at 2PM and 8PM, I do not remember exactly and do not see it mentioned at their website. You can take there private classes every day.