I guess it would be cheaper to take a bus to Arica and take a domestic flight.
Still, the cheapest would be to go overland by bus all the way, although this would take around 30-35 hours via Arica.
There is no train from La Paz itself, only one from Oruro (south of La Paz) down Bolivia to the border with Argentina.
You can however take the train, stop in Uyuni and do a tour to San Pedro de Atacama.
But as you didn't mention it, I don't know if you want to go to Santiago as quickly as possible or do you want to explore things in between?
In the latter case, there are basically 3 routes:
1. La Paz - Arica - going south to Santiago
2. La Paz - Uyuni - San Pedro - going south to Santiago
3. La Paz - Salta (argentina) - further south to Mendoza and then west to Santiago
You can also go to Salta and then take a bus to San Pedro.
Have fun!
Michael