In addition to the excellent suggestions Camlan has made, I'd also recommend:
Rustem Pasa mosque (small, but with fantastic iznik tiles - usually much quieter than the Blue Mosque)
Galata Bridge (where you can eat freshly caught fish, as you sit looking over the river)
Galata Tower (good views, provided the day is clear)
Süleymaniye Mosque (check first - it's been undergoing restoration and some parts have been closed)
Although most in Instanbul are too polite to insist, it is courteous to take a scarf to cover your head when you visit a mosque (and it is noticed, and appreciated!)
I founbf first thing in the morning was the best time to visit the Blue Mosque - it was quieter then than later in the day, with fewer groups. It and Haighia Sophia are right next to each other, and close to the Cisterns, so you could visit all three on the same day.
You can, I think, pre-book tickets for Topkapi Palace to avoid having to queue. I would recommend doing this - I visited in the middle of winter when there were not many tourists but there was still bit of a queue, and I know my sister & BiL who have been several times say it is worth booking ahead in busy times.
Be aware that the the museum at Topkapi has a number of relics relating to the Prophet Mohammed, so it is a place of pilgrimage as well as a tourist attraction. (there aren't any restrictions on how you dress, but there are likely to be visitors who will stop to say a prayer in that part of the museum, and the atmosphere of those rooms is a quiet and respectful one)
Enjoy your trip!