You have planned an amazing tour there!
Toni is right, the airport as out of the city and to the north west of it the country side is already quite empty. The highways around Santiago are quite busy, but not as busy as you would expect from a 6 million people city. It depends of course on what time you are arriving. Sundays mornings are very quiet for example. I lived in Santiago for a year and with the triathlon club we used to train on the highway between Santiago and Los Andes on Saturday mornings.
The road to La Lampa is one of your best choices to leave the city. You will easily find a place to camp. Just hop over a fence, walk 200m and hide yourself behind some bushes. Or ask a farmer to camp on his land.
See my Article about Chile: http://www.cyclocamping.com/proddetail.asp?prod=ARTI-Chile&cat=87
I have written also about Argentina and Bolivia, Look in the tips and tricks section on cyclocamping.com
The road from Arica to the East is extremely beautiful. Parque Lauca in Chile, and Volcan Sajama in Bolivia are amazing. Good choice to go there!
To give you a route suggestion between Santiago and Arica: I would cycle at least once to Argentina and another way back. The easiest pass is the one between Los Andes and Mendoza. But the Paso Agua Negra and Paso San Francisco or even more beautiful. This will also give you some experience with the very high altitude, before you go to the altiplano in Bolivia. Also the northwest of Argentina is excellent for bike touring. Chile is great too, but your only option is the Panamericana through the Atacama desert, with a few possibilities of side roads. I would definitely do part of it, but north of Antofagasta it is much more beautiful. The coastal road (number 1) from Antofagasta to Iquique is fantastic. From Santiago to Arica is almost 2,000km! And all the trucks take the same road (but north of La Serena it is not really busy).
If you want to camp as much as possible, you can easily avoid using any accommodation at all. Everywhere in Chile, Argentina and Bolivia it is very easy to find a camp spot, and it almost never rains. These countries are also relatively safe.
Have fun preparing the trip and daydreaming about deserts and huge mountains!