As cost appears to be a big issue with your trip, note that if you fly internationally into SCL and into EZE with a US passport, you will pay $131 for each country. If you enter Chile or Argentina at other points, you will not pay the fee. If you are planning to visit Brazil on the trip, you will need a Visa which costs about the same (no matter where/how you get there). Most places (if not all) now have all departure taxes included in the ticket, but maybe there are a few that don't, which can be another unexpected expense.
Sometimes the LAN website has some appealing sale fares. From LAX/JFK/MIA you can get to many points in South America with no more than 1 stop. (And via LIM you can fly into IQQ (north of Chile) or COR (Argentina) where you won't pay the entry fee, or via SCL you can fly to MDZ which is another way to get to Argentina w/o fee.
As far as safety, Chile is the safest place in South America. Note that May/June may have some marginal weather, especially as you travel further south. Don't plan on any Patagonian adventures.