Iraqi Kurdistan is generally very safe for solo female travelers, particularly in the major cities and tourist areas. The Kurdish people are incredibly hospitable and welcoming, and you'll likely encounter friendly curiosity rather than issues. Dress modestly, especially when visiting smaller towns or religious sites; long sleeves and trousers or long skirts are advisable. A headscarf isn't typically required but can be respectful in certain contexts.
For must-sees beyond Erbil, definitely explore the ancient Erbil Citadel, then head north to Duhok. From there, visit Amadiya for its stunning mountain views and historic sites, and consider a trip to Rawanduz and the Gali Ali Beg waterfall. Lalish, the holy site for Yazidis, is also a profoundly moving experience.
Transportation can be managed. Shared taxis (service cars) are common and generally safe for inter-city travel. For more remote or scenic areas, hiring a private driver for a day or two is highly recommended for convenience, safety, and flexibility, especially as a solo woman. This allows you to reach places public transport might not easily serve and gives you peace of mind.