Hey Nate and Jessie!
Ahh how exciting! I spent six months in SA in 2008 and I can tell you it was the best six months of my life - SA has so much to offer, you will have a fantastic time wherever you end up going. I went to Argentina, Chile, Peru, Bolivia and Ecuador, roughly in that order. As has already been mentioned, Bolivia, Peru and Ecuador were definitely the cheaper of the five, however you can also manage in Argentina and Chile on a budget.
I spent a month in Ecuador and was there for New Years, which was fantastic, in a little beachside town called Montanita. Prices did go up on NYE and the week afterwards, but I didn't get hit too bad because we'd gotten there just before NY. We were paying about 8 AUD a night. I think at other times of the year it's closer to 4 or 5. There were mostly tourists but there were still a lot of Ecuadorian tourists as well which made it a good mix! I would also highly recommend Banos and Vilcabamba. Try some of the candy that they make in Banos! And climb up the mountain. I spent a few days in Cuenca as well, which was nice but not as rewarding as the other two. Quito had some interesting sights (we went to this huge food market/hall which had all sorts of animals slow-roasting whole) but it didn't feel as safe as a lot of the other places I went. The people at the hostel recommended you come in as soon as it gets dark - not sure if that was a marketing ploy but I decided to keep my skin rather than find out. Guayaquil gets a lot of rap for being a big, dirty city, but it didn't seem that bad - I went to a park where there were just a whole lot of giant iguanas lazing about! The train from Riobamba was not properly functioning and it was a total waste of time going there.
Bolivia is a traveller's dream - from there you can catch a boat safari kind of thing into the Amazon (Rurrenbarque (I know I spelt that wrong)). I think it's cheaper to go here than in Brazil. La Paz is a really interesting city, you should check out the music museum, I thought that was cool. You can also book a bike ride down the 'World's Most Dangerous Road', we had a great guide called Matt and sped down it in record time. It would have been nice to do it a bit slower and really appreciate the breathtaking views but it was still nice. You also end up at an animal sanctuary for lunch. We booked that through the travel agent in the Wild Rover Backpackers, but travel agents everywhere are advertising it.
Think the hostel was about 5 dollars a night.
Lake Titicaca is definitely worth the hype - Copacabana is a pretty small town with a lot of tourists but it's obviously easy to get on the lake and everything. My favourite place was on the Peruvian side, we went to some ruins just outside of Puno. I can't remember what they were called exactly but that scenery was my favourite. We also stayed for a night (my birthday!) on an island called Amantani I think, with a woman and her three daughters who opened their home to us. They made us delicious food and gave us tea, and at nighttime they dressed us all up in the traditional Bolivian dress and taught us Bolivian dance. That was great. Visiting the little reed islands felt pretty touristy (you get the feeling they wouldn't live there if not for the tourists) but still worth it, you get to go in a reed boat and stuff.
Peru has so much to offer. I spent Xmas in a town called Mancora, I think a lot of Peruvian tourists head there around that time too. More nice hostels are springing up there now, so it's been four years, it might be way more touristy now than it was back then. Huanchaco and ChanChan have some very interesting and accessible ruins, and not that many tourists. Lots of cheap seafood. Huacachina is good for trying sandboarding on mountainous dunes, and crazy dunebuggying with a guide. Cusco and Machu Picchu are of course highly recommended. You can get really cheap food by buying the meals of the day - they give you soup and a drink and a main meal and it costs like 3 soles. So delicious too. While you're in Cusco check out the Sacred Valley ruins as well, a lot of them had a lot less tourists than Machu Picchu, and some have cool markets and stuff.
Stayed for a month in Cusco, at a hostel that cost about 10 dollars a night (that was probably one of the most expensive hostels there).
Safety wasn't really an issue - I got my camera stolen once (by a taxi driver on the way to the airport to go home!!) and an attempted thievery (had my jumper in a basket in the basket on the front of my bike, a motorcyclist drove past and took the jumper), both in Argentina but other than that had no issues. Except didn't feel safe in Quito, but may have been overly paranoid.
Only went to San Pedro de Atacama in Chile - was good but a bit pricey and all tourists.
Argentina - see Salta, Iguazu, Cordoba, Rosario (only really need a day or two for the last one). Patagonia is the expensive part of the country. You can get by on 25 USD a day in Buenos Aires (depending on what kind of hostel you choose) if you buy food at supermarkets. A lot of hostels have kitchens and stuff in them (we did this in Cusco as well).
All over though there are people selling dirtcheap street food that tastes amazing. I didn't get sick at all from eating any of it ... in fact the only time I got foodsick was the first time I cooked for myself!!
Good luck! Wherever you choose I'm sure you'll have a great time 😊