Go where the A44 crosses the A329 (location on map), and get on the A329 towards the A92. This is the motorway between Sevilla and Granada.
NORTH: A good place to start is Granada's service station ("via de servicio") in the north of the city (that's where most of the people heading north enter the motorway anyway): Where "Carretera de Jaen" (also called "Av de Juan Pablo II"?) and the A44 meet, you find a big roundabout below the motorway. Take the exit "via de servicio" (or if that exit doesn't exist, then "poligono indistrial" - sorry, not sure!) There's a gas station where you can find a ride. Have in mind that most of the drivers there will go to Jaen.
Get any of the entrances next to the A44 and use the direction of Lorca/ Guadix (A92 and later A91). Best thing would be to wait for someone who actually is going into that direction since the A44 crosses the A92 after 5-10 minutes.
Go with the bus 123 from Caleta or Estacion de Autobuses, to the first stop in Albolote, walk there about 5 mins towards the A92, which is the Motorway between Sevilla and Murcia, take a view on the right and choose the petrol station that leads you the direction you want to go. It is a really good spot, takes between 2 minutes and 8 hours. Direction Sevilla there is one more of the good gas stations, right after Lorca. The other way you have one directly when you come close to guadix, might be like exit 296.
Works not so good in direction Malaga. Direction Malaga is completely Spanish typical. That means 'Get lucky or you're screwed.'. We were 3 people and one driver could only take 2 of us so I just stayed there for the next 8 hours and nothing happened. Went back to town, took a 5 euro bus south to Motril and camped near the city as its pretty small.
Direction Málaga via Santa Fé
We walked all the way from Granada to the first gas station in Santa Fé, which is definitely not recommandable (but doable in a way). However, talk to people there and hope that the gas station guy doesn't mind. He did when we tried, but got lucky in time. It took us one car to get all the way to Málaga! (September 2010)
Granada (map) has a national motorway called A44 just around the city. There are many exits on this motorway. When you are on the A44, wait for exits with direction city center. The driver can easily get off and on the A44, in either direction.
If you talk to nice people in the Albayzin, they may be able to help direct you to some nice camping spots near the caves in the hills of Sacromonte. However, prepare for a bit of a hike with heavy gear. Anyway, you can find spots to camp outside the city on your own: from Albayzin it's not far to the woods above the city.
There is a weekly couchsurfing meeting in Granada (Winter 2009/2010 it was each Wednesday) in Granada - Time and place in Granada's couchsurfing group. You might have some luck to find a couch, just going to that meeting and get to know hosts for a night.