1. By Plane
Pegasus Air and Sun Express have nonstop flights between Antalya and Kayseri (ASR) in the summer months. These flights take a little over an hour.
The alternative is to fly on Turkish Airlines or Pegasus Air via Istanbul, with a flight time of about 3.5 hours or more.
There are flights between Antalya and Ankara. Then you must take a bus or drive (4-1/2 hours) from Ankara to Cappadocia.
2. By Bus
Overnight buses run between Antalya and Nevşehir, a 9-10 hour journey. From Nevşehir, you continue to your Cappadocian town by shuttle minibus (servis). You might also decide to take a bus to Konya, see the sights there, spend the night, and continue to your destination the next day. Here's how to buy bus tickets in Turkey.
3. By Car
The 586 km (364 mile) drive takes between 8 and 9 hours, depending on how you drive, how often you stop, and whether you're going north uphill to Konya or south down the mountains to Antalya.
It's an interesting ride, along the ancient Silk Road past the Sultanhanı and other Seljuk Turkish caravanserais to Konya (good for a meal or overnight stop), then through mountain country down to the Mediterranean coast.
Remember that rental/hired cars not returned to where you rented them are subject to a return fee, which can be substantial, to get the car back to where it was rented.
Check with one of my recommended travel agencies for flight reservations between Antalya and Cappadocia.
Here's how to go from the airport in Cappadocia to any Cappadocian town, and your Cappadocia hotel.