Stavanger is the 4th largest city in Norway.
Peregrine has heard of many travellers hitching out of Stavanger, mostly north, which couples seem to manage in 20 minutes.
North towards Bergen
An important thing to remember is that in order to go north, you are going to have to ride through on several ferries... which charge per-person fees. The likelihood of getting a completely free hitch is improbable if not unfair to the driver. That being said, there is one really good option for getting a lift north right outside the city centre, where Madlaveien hits E39 headed due north. You'll find a bus stop there where you can stand and cars can easily pull over to pick you up (use a sign)!
If you're not carrying a heavy bag, one option is to walk from the city center (or you can check the buses there) to Myrveien. There is a bus stop there and enough space for cars to pull over. People are so nice that they might even drive you up until the ferry!
South/East towards Oslo
Peregrine had not heard of anyone hitching east towards Oslo directly. Most travellers go north towards Bergen and try their luck from there. However, E39 runs from Stavanger south and then east. It is possible to get short rides within the city in order to transfer further out. At this point, E39 is still an accessible 2-lane road, so it is easy for drivers to see you and stop. If at all possible, try to go east towards Ålgård, where drivers are only going in a south-east or north-west direction. This is where Peregrine waited 2 hours on a Sunday afternoon before finally getting an English-speaking ride east at 15:00. Many drivers did stop and were friendly (people generally waved, honked horns and were supportive), but weren't going very far or didn't speak English (not a typical problem in Scandinavia). The last option drove to Mandal, which along with Kristiansund is a good option for a stop-over on the way to Oslo. Hitching out further from Mandal -> Kristiansund -> Oslo took a series of 3 hitches the following day but was straightforward. Given this information, if you manage to catch a lift to Ålgård early enough, you might make it all the way to Oslo in one day...
Accommodation and Sleep
As long as you are 150 m from buildings and not downtown, you can camp pretty much anywhere in Norway. The weather forecast site www.yr.no can tell you if there is a danger of forest fires, in case all use of open fire and also often gas stoves is prohibited.
Also, couchsurfing hosts are plentiful and helpful in Stavanger, because of the oil industry (Stavanger is the oil capitol of Norway), there are many people working shifts 3 weeks on - 2 weeks off, so it is worth a shot!
Other useful info
Many say Stavanger is the most expensive city in Norway - but if you avoid the big restaurants it is really the same as anywhere else.