Rent a car
Rent a car or even a camp van and drive up to North Iceland! The drive is takes around 3-5 hours from Keflavik / Reykjavik depending on your destination.
Please note that camping is only allowed on official camping places, both for tents and campers / Camper vans. Be prepared for a fine if you park overnight on parking lots or elsewhere in nature.
Off-road driving is also strongly prohibited by law.