It's 720 km (447 miles), a drive of nearly 12 hours, so most people going by car or bus break their journey with an overnight in Pamukkale, a lunch stop in Eğirdir, or an overnight in Konya, or all three. Here's an itinerary.
You may find a direct bus all the way from Nevşehir(Cappadocia) to İzmir, but it's also possible you'll have to change buses enroute—not a big problem. From İzmir there are frequent minibuses south to Selçuk, Ephesus and Kuşadası, and regional trains to Selçuk.
If you plan to spend a night in Konya or Pamukkale along the way, you may have direct buses for two or all three segments of your journey (that is, from Izmir to Pamukkale, from Pamukkale to Konya, and from Konya to Nevşehir.) Here's how to buy bus tickets in Turkey.
Most travelers choose to drive via the Meander River valley, from Selçuk (Ephesus) via Aydın to Denizli and Pamukkale, so they can detour to ancient Aphrodisias along the way, then spend the night at Pamukkale. This makes a good first day of travel (190 km, 118 miles, about 3 hours, not counting the detour to Aphrodisias).
On the second day, it's 306 km (190 miles, 5 to 6 hours) from Pamukkale to Konya through Turkey's lake country. Stop at Eğirdir for lunch, and enjoy the scenery. Stay the night at Konya to see its great Seljuk Turkish buildings (especially the Mevlana [Rumi] Museum). The next day, continue east along the ancient Silk Road via Sultanhanı to Nevşehir, Cappadocia(221 km, 137 miles, 3 hours) on Day 3.
For details, see my 5-Day Ephesus—Cappadocia Driving Itinerary.
By Train & Bus
If you're not in a great hurry, one of my favorite ways to do this journey is by the Konya Mavi Tren, a nightly sleeper train from İzmir to Konya, then bus or rental car from Konya along the ancient Silk Road to Cappadocia.
You board the train in the evening, dine aboard if you wish, bed down in your private sleeping compartment, breakfast on the train the next morning, and arrive in Konya before 08:00 am. A quick taxi ride (10 minutes, TL10) takes you from the Konya Gar to the Otogar (bus terminal), though you really should spend most of the day seeing the sights in Konya, then take a bus eastward to Nevşehir in the late afternoon. More...
Another train-&-bus combo, with longer train and bus rides, is the İzmir Mavi Tren (a sleeper train from İzmir to Ankara), then a taxi from the Ankara Gar to Ankara's AŞTİ main bus terminal, where you buy a ticket for a bus from Ankara to Cappadocia, a 4-1/2-hour ride to Nevşehir. More...