The ring road can be boring and also dangerous to hike. There are strong winds in the south including a lot of ashes or sand that sometimes even polishes cars and crashing windows. When we drove the ring road in the south the last time (August 2013), the wind was really really heavy and I was glad to have a car.
I would recommend to hike in the highlands. Choose a route according to your experience, equipment and time of the year. You will see much more of the amazing nature.
I'd like to emphasize again that the weather in the highlands (and all over the island) can change very quick and you should be aware of this when planning your route. There is an authority where you can register your route to get help if you get lost: Safetravel
There are also emergency shelters all over the island. Make sure you know where the next shelter is before starting the day.
I would recommend to take a mobile phone and get a prepaid card from Simmin and you will have at least GSM all over the island.
Camping is generally allowed in Iceland but you should ask the owner of the land/property where you want to stay. There are also a lot of camping places where you have a shared washing room and kitchen. I think they are quite cheap.
Personally I found Askja especially interesting. It's a bit like hiking on the moon (without marked trails). There are some shelters were you can spend the night or sleep in your tent. If you go there by car make sure you know how to cross a river.
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