Yes, I think you can manage Turkey by yourselves, not in a packaged tour. The people are enormously friendly and helpful -- they'll take you by the hand and lead you where you need to go, for free, whereas in Italy, if they do that, they'll ask for 20 Euro for 15 minutes of assistance.
You might want to join a tour of Istanbul that picks you up from your hotel, as there is so much to see. My siblings did that and had a great time. I can send you their tour recommendations.
If you plan right, you can get roundtrip airfare for less than $1,000. (One friend got a round-trip ticket for about $700.) You can fly from Istanbul to Kayseri (the stepping off point for Cappadocia) for $50 using a domestic carrier, then take a bus from Kayseri to Goreme, Cappadocia for $14 or less RT (90 minutes on bus). Or, there are express shuttle buses from the Kayseri airport to Goreme, Cappadocia, for 10 Euros. Great hotels in Cappadocia are $100 or less. We've stayed at the Kelebek (meaning "butterfly"), which operates shuttle bus from Kayseri airport. If you come on weekend to Kayseri, we can meet you and get you to Cappadocia, recommend a "cave hotel" and our favorite restaurant with live music and "pottery kabobs."
Istanbul and Cappadocia are "must sees." You need a minimum of three full days in Istanbul, preferably 4 or 5; and a minimum of 2 full days in Cappadocia (preferably 3). More if you really like hiking. You could cut your costs if you just go to those two places.
There's Ephesus if you like ruins and ancient history. You'd have to fly there from Kayseri by Sun Express, the cheap domestic carrier, maybe as low as $45 one way (you'd fly to Izmir).
Three hours from Izmir/Ephesus (by bus, less than $20) is Pammakale ("Cotton Castle"), an ancient mineral spring that's heated naturally, with calcium deposits that look up a hill of cotton or snow.
Southeast Turkey is fascinating, though dusty and impoverished in places. There's Mt. Nimrut (with the god-like statues on top), Gaziantep (with an incredible museum of Roman mosaics and a very classy Ottoman-era restaurant), Sanliurfa (home of Abraham and Job), and a "sacred fish lake" -- very Arabic atmosphere. You'd probably have more fun doing this leg of the trip with a group in a small bus. That's what we did.
Turkey strikes me as safer than the US -- the murder rate is a lot lower than U.S., and I never hear of street crime. Maybe very occasionally in Istanbul.
In short, I can't recommend Turkey more highly. I think you're in for a great adventure.
I am so thrilled you are coming. I wanted to make some suggestions, so you could start figuring out your itinerary. For any domestic flights you take (and you'll want to because the Turks are lethal drivers, buses and trains are possible but trips are l-o-n-g), the earlier you book, the cheaper the flights will be. I can help you with those flights because I know the carriers, etc.
ride an hour to Cirali. The beaches are like beaches you'd never see in the
States, pristine and bordered by ancient ruins; we stayed at a resort with tea
time, massages under "sheik"-like tents by the pool, and other glories. At
night, you can walk about 5.5 km roundtrip to see the chimara on Mt.