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There is only one road out of town in this direction, but the problem is that there is very little traffic (at least that was so in April, 2008; expect a similar level unless in summer), so be ready to wait for several hours. Also, if you are around between April and October, remember to take lots of water as well before setting out from the town.
The roads are quite curvy, sometimes extremely so, until Alanya, which is several hundred kilometres away, so those with motion sickness may want to note that. Fortunately there are some towns on the way where you can stop to take a break yourself.
Personal experience: Theo hitchhiked to Alanya and back two times on this road and although it is scarcely trafficed it is well worth due to the experience as this is one of the most beautiful roads in Turkey. I'm normally not a big fan of tourist sites but Anamur castle (5 kilometers or so before the town) is well worth a stop. It is also possible to camp inside the castle or sleep in one of the towers. Just give the guard a smile and a locally grown banana and he'll probably give you his permission. If he doesn't? Just do it anyway. You will not regret it.
There is a wonderful small tranquil campsite by the sea about 2 kilometers east of the town. One tent should be 10 lira a night and probably slightly more in July/August.