I went to Istanbul with friends in January 2014. Due to a lack of preparation we ended up only visiting 2 of the 4 inscribed components. The first was the beautiful Süleymaniye Mosque. A prayer had just finished, so we went up to the entrance, took off our shoes and went inside. You don’t have to pay to get into the “working mosques”, and it is well worth doing – don’t just go to the inactive Hagia Sofia! Inside the mosque the floor is covered in a rich carpet, and the walls and ceilings are exquisitely detailed.
It was a nice day outside, so after the Süleymaniye we went for a beer on a rooftop café we had spotted from the gardens. Except we didn’t get a beer. That’s one thing we learned about Istanbul – it’s very difficult to buy a beer within sight of a mosque. Which, in a city as richly endowed with them as Istanbul, is a lot of places!
The other subsite we went to was the Archaeological Park's museum.
We also visited the Basilica Cistern, the Galata Tower, the Hagia Sofia and the Blue Mosque.
I suppose we also saw the 'Land Walls of Istanbul', but only whilst driving past, so I don't count it as 'visited' properly.