There are lots of taxis to get around Erbil. which shouldnt cost alot , so bargin before you get in . the first taxi we got in from the airport to city center took 20.000 Iraqi derhams ,where as when we left to the airport the taxi had taken us from Familly mall , waited for us infront of the hotel for 15 minutes or so before taking us to the airport and only took 15.000 ID.
To Slemani: The direct route to Slemani passes briefly into Arab Iraq and skirts the city of Kirkuk, not somewhere you're advised to linger at present. I'm told it is perfectly safe, although it is a bit nerve-wracking when you reach a checkpoint manned by US soldiers in full body armour. I wasn't particularly keen to take this route, but asking a taxi driver in Erbil to take me to the Koya Garage led to questions about why I wanted to go to Koya...I explained I'd heard the Kirkuk route was unsafe, but he laughed and took me to the Slemani Garage instead. The route doesn't actually go into Kirkuk itself, but passes close enough to get a good look, before re-entering Kurdish territory about 15 minutes later. If you take this route, choose a shared taxi over a bus, as buses might enter Kirkuk to pick up more passengers. 15,000 will get you to Slemani in 2-3 hours, dropping you off at a roadside taxi rank miles from anywhere central.
The alternative route is to take a shared taxi to Koya for 10,000, and change for a Slemani-bound taxi. I did the trip in reverse, and the section between Koya and Dukan is especially beautiful, passing Lake Dukan and crossing a mountain pass...however, I did have to wait a while for enough passengers to fill the second taxi.
To Dohuk, ask for Karaj Dohuk, and make sure your taxi will be taking Tariiq Ba'atari, going via Baderash and Ain Sifni rather than passing through the outskirts of dangerous Mosul.
For places to the north (Akre, Shaqlawa, Haji Omaran, etc.), head to Karaj ash-Shimaal, which is about ten minutes' walk north of the citadel (take a taxi if you have luggage, or be prepared to ask everyone you pass for directions, as it isn't obvious). I waited ages for passengers going to Akre, which isn't a popular destination it seems...it should cost 10,000 per person, but as I was the only one, I managed to negotiate a ride for 25,000 on my own, with the assurance that the driver would try to pick up passengers en route.
Taxi are plenty in Erbil and you should not wait too long before one stops to pick you up.
If you take them from the street, the average price for a ride is about 1500 IRD (1 usd) as a foreigner, they may ask double. In any case, do not pay more than 3000.
The journey from Erbil Intl Hotel to Khanzad (out of town) is around 15,000 IRD.
The best way to get to arbil wll perhaps be to fly to bagdad, and then travel to arbil by car or bus. I am writing this text 2 weeks after the end of the iraq war, so I can't say what will be the best way.