Getting to Erbil, you have two options...the quick, dull and risky route...or the slow, mountainous and safe route.
1 - the quick, dull and risky route. If you're in a hurry, don't like twisty mountain roads, or have a morbid fascination for visiting more dangerous parts of Iraq, this is the choice for you. Take any shared taxi heading to Erbil (Hewler in Kurdish), and you'll pass briefly into Arab Iraq into the outskirts of Kirkuk. The first thing you deal with is a checkpoint manned by US marines in full body armour, who will no doubt want to ask you a thing or two about what the hell you're doing there. But don't worry, as soon as they know you're not stopping in Kirkuk and are heading straight along the ringroad and back into Kurdish territory, they'll let you carry on...so it can't be that risky. The section near Kirkuk only lasts about 15 minutes, but it seems much longer. Buses on this route do apparently go into the city centre, so unless you're particularly foolhardy, stick to the shared taxis.
2 - the slow, mountainous and safe route. First take a shared taxi to Koya, which travels past Dukan and the "Kurdish Seaside" before heading up and over the mountains to the small city of Koya. Views are spectacular, roads are steep, corners tight, and drivers faster than they should be on such roads, so take some travel sickness pills if you're prone to vomiting on fairground rides. In Koya, I had to wait an hour and a half for enough passengers to fill an Erbil-bound taxi, which explains why this route is the slow option.
No doubt you'll want to visit Halabja while you're here, and it's an easy day trip, assuming you leave early enough. Take a taxi to Karaj Halabcha a few kilometres from the bazaar, where you'll find buses heading to Halabja. There are two Halabjas, one where the attacks happened, and a new one built some distance away, so you'll need to ask for Halabcha Shaheed. The trip takes a couple of hours, and you'll see the big monument to the chemical victims on the right just before you enter town.
In Sulemaniye, the way to/from the airport can be tricky as sometimes there are no taxi in stand-by.
If you feel there is not enough taxi, do not hesitate to ask some people to share the ride with them.
The price to/from airport should be between 15 to 20,000 IRD (10 to 13 USD)
Between different town in Kurdistan, you can hire a taxi for the journey. It is not cheap but at least it is reliable.
I usually ask the hotel to arrange that as I feel much safer because the hotel staff know the driver.
The ride between Erbil and Sulemaniye costs about 80 USD for a whole car everything included.