The fastest and most comfortable way of getting to Cappadocia is by plane. There are two main airports that you can fly to, Kayseri and Nevşehir.
Kayseri Erkilet Airport (ASR) is located in Kayseri, about 75 km from the center of Cappadocia. Turkish Airlines (http://www.turkishairlines.com/) operates several direct (nonstop) flights from Istanbul Atatürk Airport (IST) to Kayseri. Pegasus Airlines (http://www.flypgs.com/en/default.aspx) also has direct flights from Sabiha Gökçen Airport (SAW) on the Asian side in Istanbul. There are also direct flights by Pegasus Airlines and Sunexpress (http://www.sunexpress.com/) from Izmir to Kayseri a couple of times a week and daily flights via Istanbul. During the high season Pegasus Airlines also has direct flights between Antalya and Kayseri a couple of times a week.
Nevşehir Kapadokya Airport (NAV) is located in Gülşehir, about 40 km from the centre of Cappadocia. Turkish Airlines operates direct (nonstop) flight from Istanbul Atatürk to Nevsehir Kapadokya twice a day.
All domestic flights arriving at or departing from Cappadocia (both Kayseri and Nevşehir airports) have regular transfer service available, however, advance booking is required and the drivers cannot accept passengers not on their list. It takes about 1.5 hours to get from Kayseri and about 1 hour from Nevşehir airport to Göreme.
When leaving Cappadocia by plane, you will be picked up from your hotel approximately 2.5 hours for Kayseri and 2 hours for Nevşehir prior to the departure time. You will arrive at the airport at least 1 hour before your plane leaves, allowing plenty of time for check-in.
There are 3 bus companies, which have direct (mostly overnight) buses between Göreme and major cities of Turkey every day:
Süha Turizm (http://www.suhaturizm.com.tr/),
Metro Turizm (https://online.metroturizm.com.tr/BuyTicket.aspx),
Nevşehir Seyahat (http://www.nevsehirlilerseyahat.com.tr/)
Approximate duration of a bus journey to Göreme from:
Istanbul 11 hours, Ankara 5 hours, Konya 4 hours, Adana 5 hours, Pamukkale 10 hours, Kuşadası 13 hours, Kayseri 1.5 hours, Fethiye 12 hours, Bodrum 14 hours, Izmir 13 hours, Bursa 10 hours, Antalya 9 hours, Çanakkale 17 hours, Trabzon 11.5 Hours, Marmaris 14 hours