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.
Take the bus SN3 or SN4 and get out at Faculty of Fine Arts. (Facultad de Bellas Artes) From there, continue walking straight on in the same direction for about 15 minutes and you'll get to a petrol station Repsol. The traffic is low and it's 8 km before the highway. Try to get a ride after Santa Fe.
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 petrol 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.
Take the metro to the last stop "Albolote", then walk north towards the highway (ca. 2-3 km). There is a gas station on the left handside of the bridge over the highway. It is a really good spot. Direction Sevilla there is one more of the good petrol stations, right after Lorca.
Direction Málaga via Santa Fé
We walked all the way from Granada to the first petrol station in Santa Fé, which is definitely not recommandable (but doable in a way).
From the end of Granada you can hitchhike a car to Santa Fé.
In Santa Fé you should check really hot springs!!
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.
Just after the district Albaicin there is mountains where you can find good places for a tent. It's about 30 minutes on foot from the city center or from Alhambra. Locals who live there do not care so much about tourists with a tents. Also in this hills there are some caves where people live. Maybe you'll be lucky to find any empty cave for your own or someone invite you.