This one is easy. The main mobile providers in Chile are Entel, Claro and Movistar. Chile uses the same GSM frequency as the US, Canada, and most of Central and South-western America so if your phone is from one of these areas, it will work. If not, a cheap handset costs around CLP$ 15.000.
Pre-paid plans are easy. You can find Entel, Claro or Movistar stores within five minutes walk of almost anywhere in Santiago. Ask for pre-pagado. The clerk may ask you for a RUT when you purchase the phone. If you have one, feel free to give it and your phone number will be linked to that RUT, this can be useful as it allows transfer of numbers when you get a new phone.
Protip: For gringos who need a data plan, your absolute best bet for a cellphone service is the Virgin Antiplan. You cannot get a standard contract without a permanent residency, and the Antiplan offers the best data packages for a pre-paid plan.
If you don’t care about data and use your phone little, Entel or Movistar pre-paid plans are fine.
You can recharge pre-paid plans almost anywhere including at convenience stores and in the metro.
If you don’t have a RUT, it is not compulsory to give one. If pressured, you can just give your passport number or even make one up. There are also places where you can recharge your credit all over the city.
Contract phone or internet plans are more difficult and require at least temporary residency, Chilean bank account and a carnét with valid RUT number.