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Welcome to the forum Vaaai.Yes,you can expect snow for the full length of the route but this would be driving along major highways almost the whole way and these are generally kept open and clear.But it still wouldn't be a relaxed 10 hour drive,snow falls,ice and drifting freezing fog are other hazards you may encounter.
In addition,unless there are 3 or 4 of you to share the costs it will be expensive,we have the worlds most expensive fuel here in Turkey :( and this is on top of the hire charge for the car.If there is a group of you then yes,it will probably be more economical but if you are solo or there are just two of you then consider either taking the overnight bus from Istanbul or flying instead.
Domestic flights from either of Istanbuls airports are often very reasonable and you can go into either Nevşehir or Kayseri airports.If you choose this then try to get flights from Istanbul Ataturk airport not SAW airport which w-a-a-y out on the Asian side a long way from the city centre!
Carriers include Turkish Airlines and Pegasus.
Unfortunately there are no trains at the moment as the line from Istanbul is closed for up date and repairs until 2015.

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Fly...or take a overnight bus...then rent a car once in the area. And those cave hotels are freezing that time of year, as people I met 2 weeks ago when I was in Istanbul had returned and were not comfy at all...

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Haa yes,there are still a few hotels around Cappadocia that aren't set up with decent heating systems.This is an important point many of us mention to winter visitors;make sure the hotel you choose has a proper functioning heating and hot water system and doesn't just rely on solar power and a log fire.The fire may look lovely and romantic and is warm when you sit by it but it doesn't project enough heat to properly warm the rooms-and a cold shower is not pleasant!

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Thank you very much, guys. It was very helpful :)