As a general rule, I would advise against renting a car for the entire trip. Renting a car in Turkey is expensive. The bus network in Turkey is very dense. Bus travel is comfortable and not expensive. Prefer the busses for long distance travel. You may consider to rent a car locally to access some sights. For very long distances, flying may be an option too. It's more expensive than taking the bus, but you save a lot of (precious) holiday time. Don't forget that Turkey is a big country and distances can be very long.
It would be a shame not to stay some days in Istanbul. It is a great city not to be missed. Furthermore it is interesting to observe the contrast with the rest of the country. Cappadocia is a pretty unusual natural place. It should be on you itinerary.
For the rest it depends on your style:
If you want a packed trip, you can consider to go from Istanbul to Ankara and then to Cappadocia. Next you could go to the Mount Nemrut (Nemrut Dağı), another "unusual" place. Finish you trip in Şanlıurfa and the go back to Istanbul, possibly by plane.
If you prefer a trip with less kilometers, stay some days in Istanbul, travel then to Bursa. It's an interesting city, a bit off the beaten track. Then go to Cappadocia, via Ankara.
If you really want to do it in depth, split your time between Istanbul and Cappadocia.
In July, I would stay away from the Mediterranean, unless you do have to go to the Sea. It's really crowded then. If you do want to go to the Mediterranean, go to Çıralı, some 80 kilometers West from Antalya.
If you are tempted by mountains, steppe and austere landscapes, you can consider a journey in the East. Contrary to what I say above, I would recommend that you rent a car for this precise trip. It goes through sparsely populated regions, where the bus network is not as good as it is in the West. It's possible to do this trip on public transport, but a private car is the better option.
Fly from Istanbul to Erzurum. From there you can make the following tour: Eruzururm - Yusufeli - Kars - Doğubeyazıt - Erzurum. That keeps you busy for a week or even more. If you like slow travel, you might also consider to replace the flight to or from Erzurum by a train journey. The Doğu Ekspresi is a daily sleeper train running between Ankara and Erzurum. It takes +- 16 hours. Combined with some days in Istanbul, and possibly a day in Anakra, the 13 days are quickly filled.
Have a look at Turkeytravelplanner.com. It's a very good and quite complete resource for traveling in Turkey. All the places I mention are described there.