I came back home from wonderful Istanbul just three hours ago, and I thought first thing I'd write a quick post to thank you all for the valuable information I received here, which allowed me and my parents to have a great time during our trip (don't worry, I'm not too tired; it was just a 2,15 hours flight, with no jet-lag); it was actually one of our best holidays ever.
I'll write an extensive trip report in the following days (be prepared: it's going to be very long and detailed!), but what I'd like to remark now - in case the "safety concern" issue is still around - is that we never felt unsafe for a second, in any place we visited; we loved, instead, mingling with Turks in off-the-beaten-path places where we seemed to be the only tourists around, and were very impressed with how polite and friendly people were (I mean: common people: sellers and restaurants touts, sometimes, bothered us a little bit).
Oh, and we loved the food! (Note that we're Italian, so we know our stuff when it comes to food!)