La Serena is a city in northern Chile.
East towards Vicuña, Valley Elqui, Argentina (Ruta 41)
From "Plaza de Armas", go West (opposite of the church). Continue down for 650 meters to the Mall Puerta del Mar (there is a Lider and a Sodimac there). Before you get to the pedestrian bridge that crosses the big road to the Mall, there are the buses that go to Vicuña / Pisco Elqui. Take any of them and ask to be dropped of at the airport. It should cost you 800 pesos (as of June 2016). Eristoff got picked up in 20 minutes, 10 minutes more down the road.
Argentina can be reached by Paso Agua Negra (4775 m), on a fully-paved road up to Rivadavia, then partly-paved.
South towards Santiago (Ruta 5)
Take a bus going to Coquimbo which can be caught anywhere along Ruta 5. Cost is 600 pesos or less depending on what poInt of the city you board.
You could then probably catch another local bus to take you further along Ruta 5 as there are more bus stops, or you can just start walking (roughly 4.5 kilometers from the Lider).
You will see a couple decent sized pull offs which are bus stops but don't stop there. Keep going until the road begins to go straight uphill for a ways. At the top of this hill there is massive amounts of space for cars and trucks to stop and some will already be going a bit slower having just climbed the long hill. Also it appears that several trucks stop here anyways for various reasons.
Accommodation and Sleep
There are gas stations next to the bus terminal which tolerate camping hitchhikers. Alternatively, hitchwiki user Eristoff had some success sleeping on the dry river bed of Rio Elqui (from "Plaza de Armas", face the church and go left, you will find it in about 700m).