Kayseri and Nevşehir are both outside (to the E and W respectively) of the true Cappadocia area.
Göreme, Ürgüp and Avanos are the main (small) towns of Cappadocia but there are other villages with accommodation with Avanos being the largest and most "Turkish" but also with the least accommodation.
For the short time you have to spend and given the time of year I suggest Göreme unless perhaps you want to splurge for a really nice place at a very off-season price in (nearly as convenient) Ürgüp.
While you can certainly just arrive and find accommodation I do recommend that you make advance reservations and also arrange for convenient and inexpensive transportation from the airport (most likely Kayseri) to your hotel. I also suggest that you pre-arrange with your hotel for the balloon tour and any other tour you may want to take. The people and businesses in the Cappadocia area are very fair in their business. Just be certain that you are dealing DIRECTLY with the hotel/hostel/etc. and you can be nearly certain that the prices you receive are as good or better than if you do things on your own and more importantly you won't waste time upon arrival finding accommodation and arranging a balloon flight and/or tour(s)!!
On the subject of reservations, you should ALWAYS reserve Istanbul accommodation in advance!
