What's an easy, inexpensive way to get to Göreme from Ankara?
Metro bus company has a bus everyday at 15:30. It takes over 4 hours and costs 40 liras (=$11). That might seem suspiciously cheap, but don't worry. Buses here offer much higher quality and comfort (usually including a steward serving snacks, free wifi, movies, etc.), compared to those in the US (such as Greyhound or the dreaded Chinese bus that runs between NY and Boston).
You can (probably) find other more localized companies in Ankara's bus terminal, but Metro is a nationwide reliable company, so if its hours fit your schedule, no need to search for an alternative.