Outside the airport terminal there are taxis that will take you into the city. You would ask to be taken to Garage Diana (pronounced dee-a-na) which is close to the centre and this will cost approximately 5,000 IQD. From the garage you take a taxi to Diana/Soran or a minibus. The taxi costs 40,000 IQD for one person, but if shared with others this reduces to 10,000 IQD per person. A minibus will cost you 6,000 IQD.
The journey takes approximately 2 hours through beautiful mountainous countryside.