A couple of notes on Lima and Santiago follow.

Lima: airport is in Callao, the run-down port district. There's not a lot to see or do there. You will likely end up staying somewhere in Miraflores, there are a bunch of hostel-type places in the vicinity of Parque Kennedy. I'd head there from the airport and look around. You can get cabs from the airport for approximately US$40 (for a secure "remisse") or S/.40 (out front of the terminal, be sure to negotiate the fare before you get in). Be careful with public transit in Lima, it's extremely cheap but can be risky, and the route map, if one existed, would be a spaghetti bowl of confusion.

Santiago: I have no idea where the cheap places to stay are, perhaps in the center of the city? Airport is in Pudahuel, WNW of the city. Best and cheapest way to get into town is to catch the blue bus that leaves from in front of the international arrivals area every 15 minutes to the Pajaritos metro (subway) station. The bus fare is CLP$1.400 and the metro fare is approximately CLP$600 depending on time of day. Public transit is generally safe in Santiago and in particular the metro system is world class, albeit overcrowded during rush hour. Taxis are generally safe, metered and cheap although not as cheap as Lima.

Have fun.