There are frequent bus connections to Antofagasta (2 hours). There are also frequent overnight buses arriving from Iquique...arriving normally in the early morning hours. The bus company will let you sleep on the bus for a few hours until morning so there is no need to book a hotel or hostel. Be careful of your bags in the bus station here, bag thieves were operating here on a daily basis in October 2012.
If the nearby salt plains appeal, take the 3 or 4 day train ride to Bolivia (bring warm clothes!). The amazing train is essentially a cargo train that carries a handful of Bolivian peasants between the coast and the altiplano, but it provides an incredible vista.
The train to Bolivia used to run 2 times a week.
As of June 2009, the train service stopped running to Ollague. However, there there are bus services to Ollague (the Chilean-Bolivian border) for around 4,000 Pesos. Try the bus service 'Frontera Norte'
Edited info: This train service has stopped. There are no trains from Bolivia to Calama, Chile. The FCA website takes you to train service from Lima to Huamcayo in Peru.
Calama is not large and most of the city centre can be covered on foot. Collectivos (shared taxis) run fixed routes and cost Ch$500, while buses, named with letters of the alphabet, cost about Ch$300.
Get a taxi in the city centre to Chuquicamata. It is an enormous mine open to the sky. There is a free tour to the mine, its worth visiting.
To see the mine you should first sign up at the local municipal tourism office. It's located a few blocks from the main plaza on Calle La Torre between Calle Vicuna Mckenna and Pedro Leon Gallo (more towards Mckenna and on your left hand side as you walk from the plaza). It's in a small building on a market-lined street so pay close attention or you'll walk right by it. They'll take your name and information and direct you to the taxi/collectivo stand that can take you to the mine as of 13:15 PM. The cost of the collectivo is 1500 CLP each way. The tour starts when a bus picks you up near the fire station and ends about 60-90 minutes later. The collectivo is waiting for you at Chuqui when the tour ends. The tour itself is free but you will be asked for a voluntary donation for the children's fund.
There's a small amusement park for children off the main square in town.
And you can go to the mall. There's a movie theater there, but be warned, all the movies are in Spanish.
Rent a 4x4 truck and go see Chiu-Chiu, Lasana, and the geysers de Tatio. For around 40,000 pesos, you can rent a 4x4 truck. Chiu-Chiu is about 35km North-East of Calama. There are intresting ruins, a lagoon, and some old churches. Lasana is a beautiful canyon winding through the desert. It's green and really different. Then, a several hour drive across the salt flats and through some VERY rugged roads brings you to the Geysers of Tatio. If you camp on the NORTH side of the geysers, and come down in the morning, you can get in for free (otherwise around 16,000 for a truck and four people). The geysers are at about 4,200 meters, and you hit abou 4,700 meters coming in from the north. In May it was VERY cold (our water bottles froze that night solid). A very cool experience.
Be vigilant about your stuff and never lose sight of any of your bags in public places like the bus station (for instance, waiting for a connection). The thieves work in groups and usually one will try to distract you while another grabs the bag when you've turned around.
San Pedro de Atacama is two hours away by bus.
The airport is a few miles south of the city. You can call for an airport shuttle to pick you up at 98162091 for Ch$3.000. An ordinary taxi is said to cost Ch$6.000. You can also consider to take a collective to the southern suburb, and walk or hitch hike the last km to the airport.