There's no train. There are three ways: fly (80 min, recommended), overnight bus (11 hrs, cheap), or drive (7 hrs, only if you love road trips).
Turkish Airlines and Pegasus run 4–6 daily flights to either Kayseri (ASR) or Nevşehir (NAV). Flight time: 80 min. Cost: $60–120 one-way, less if booked 3+ weeks out.
Kayseri vs. Nevşehir: Kayseri has more flight options and only 30 min more drive to Göreme. Most travelers fly to Kayseri.
Airport to cave hotel: Pre-book a transfer ($35–50) or take the shuttle van ($15/person, 60–90 min with stops). Kayseri airport taxis sometimes quote $100+; insist on meter or walk away.
Metro Turizm, Kamil Koç, and Nevşehir Seyahat run overnight buses from Istanbul's Esenler terminal to Nevşehir/Göreme. 11–12 hours, $25–45, fully reclining seats on premium buses.
Reality check: it saves you a hotel night, but you arrive at 06:30 tired. Balloon flight is that morning if you're on a tight plan — not ideal.
7 hours on smooth toll highway (O-21). Rent in Istanbul, drop in Nevşehir. Only worth it if you want to stop in Ankara (Anıtkabir) or Safranbolu along the way.
Most travelers fly back to Istanbul for their international flight, or continue to Antalya / Izmir. All three have direct flights from Kayseri/Nevşehir.
Activate these from home — cheaper and simpler than sorting them at the airport.