Turkey

You could go to Sinop up on the Black Sea. Its very, very old and is an electronic intelligence gathering station still in use.

My vote is to go to the Cappadocia region. The natual landscape and historic sites are unique in the world - think the badlands of the Dakotas coupled with churches built in the 800's to 1000's and underground cities that are even older. There are a number of small towns and mid-sized cities in or nearby to choose. Goreme is the classic choice. There are tons of options for your second choice. The western coast is dense with options. You could do more in central Turkey (e.g. Konya). You could go to the Lake Van area in the east or north to the Black Sea. What are you most interested in seeing and doing (e.g. history, culture, food, natural areas)? In my opnion the best way (outside of Istanbul) to travel is via air with rental cars. Flights between cities in Turkey are inexpensive. Rental cars give you flexability to see lots of smaller sites and areas not well served by buses. Please note that most cars are manual transmission and you will need to request an automatic if that's important.

Two ideas (although you've given very limited/no info about your interests): Bursa (not far from Istanbul)- can get there by ferry/bus combination
Capadoccia region - best to fly to save time Depends what you're looking for...coastline? antiquities? small towns? big city? natural wonders? If you're more specific, perhaps you'll get better tailored responses.

Without knowing what interests you there is no way to give any solid suggestions but Edirne and Galipoli (Çanakkale) are close to Istanbul. If the distances involved are 8 hours or less I suggest using the bus, which are fast and comfortable as well as being inexpensive. There are dozens of Turkish bus companies with Metro being probably the largest and best known, though not my favorite.

I second Rob's post. I have spent time in Istanbul and have taken RS Turkey tour and I agree with him.