What is the best way for a college student on a budget to travel around South America?

Some years ago I spent about six months travelling around South America. I started by arriving by ship in Rio de Janiero , Brasil then went to Bolivia , Argentina , Uruguay then back to Argentina and flew out from Buenos Aires. During those six months I travelled either by bus train or boat and only flew when I left. There is a good network of affordable buses all over South America and also regular passenger boats going up and down rivers. I travelled by boat all the way up the Amazon and other rivers from the mouth of the Amazon into Bolivia sleeping on the deck alongside the locals. In Brazil Portuguese is spoken but everywhere else it is Spanish , you can just get by with English but it isn’t easy , so learn some Spanish if you don’t have it already. I stayed in cheap hotels which you find in every town usually near the bus station but otherwise ask the locals where they are. Go to a book shop now and check our different guide books and choose one that you like. I didn’t use a guide book but it would have been easier if I had. Speak to a Doctor or Pharmacist about malaria and other disease prevention and follow his advice , South America is thick with mossies

I recommend you highly Ecuador, because it´s unexpensive and because of its small size you can see a lot kinds of landscapes (beach, mountains, volcanos, jungle, islands), unique flora and fauna, historical monuments and culture offer, very friendly people and you can eat very well. The best they use the dollar like currency.
Good luck in your adventure ;)

I recommend Ecuador and Peru. They are both affordable and almost every place has a spectacular view of something and so much to do. I left the countries having seen a lot and feeling like I missed everything. Travel by bus and stay in hostels.