In Erbil there is no public bus. There are private shared buses that travel popular routes. I have been on one or two but try to not get on if it is all men as I am a woman.
Usually I take a taxi. Local women only use taxi drivers they know and they use the same ones over and over again. Fares range from 3000 ID to 9000ID depending upon how far and how long to get there. There are no meters in the taxis. Locals just pay the fare they know which is much less than what expats pay. If you take a taxi in front of a mall like Family Mall or Majidi Mall and the taxi has the name of the mall painted on it you will pay a lot more than on another taxi. It can be cheaper to walk away from the mall a little and then flag down a taxi that is passing by. Passing taxis may honk at you. It means they are free to take a passenger.
There is Hello Taxi, expensive but good, which I use to go to and from the airport. I use this one if I have to go somewhere alone at night. I try to stay home at night unless I am with a group.
There is also now a "Pink Taxi" which is driven only by women and only for women. I do not use it because I already have a local driver I am happy with.