What a conundrum! What nationality are you that need so many visas?
Yes, you can get to Brazil through Argentina. Indeed your route would be through Antofagasta, Salta, across the Chaco to Resistencia/Corrientes (I prefer Corrientes to Resistencia as it has more character). If you prefer not to go through Paraguay, there is a direct bus from Corrientes to Puerto Iguazu. You can get your Brazilian visa quite easily here, then cross over to Foz do Iguaçu, and you are off and running into Southern Brazil.
There are excellent bus services throughout Latin America.
Buses Géminis goes to Salta and Jujuy, Argentina from Antofagasta on Wednesday and Sunday at 7:00am (US$38, 14 hours). Tickets are sold on: Tuesday, Wednesday, Saturday and Sunday. It also goes through San Pedro de Atacama on the way, so you may wish to pick it up there.
Daily buses from Salta to Corrientes, mostly overnight sleeper buses, about 13 hours US$ 35.
Daily buses from Corrientes to Puerto Iguazú. About 5 hours, US$ 15.
Skim our own Travelpod travellers who have done this route:http://www.travelpod.com/s/antofagastahttp://www.travelpod.com/s/saltahttp://www.travelpod.com/s/corrientes%2C+misioneshttp://www.travelpod.com/s/puerto+iguazu
Here's a great site for the section Corrientes to Puerto Iguazuhttp://www.brazilmax.com/news.cfm/tborigem/pl_south/id/17
All schedules are of course subject to change. Recommend you purchase onward tickets at each place as soon after arrival as possible.
Hope this helps