That, my friend, is nobody’s business but the Turks’!
Lyrics and chord progression here.
Massively seconded. It’s one of the most fascinating buildings I’ve ever been in. Standing in between it and the Blue Mosque is a hell of a sight.
An interesting point is that despite its exoticism (in our eyes), it was the Byzantine architecture of the church that influenced subsequent mosque building in the region. When you see the skyline of Istanbul with all those distinctive Muslim domes, they’re actually aping an original Christian design.
Also worth a few hours is the Basilican Cistern, which is both atmospheric as anything, but fascinating in the manner in which its construction materials were recycled from so many eras - I love the upside-down Medusa head.
The Turks are generally a pretty friendly people. You really should see Topkapi. It is very close to the Hagia Sofia and the Blue Mosque. I found it far more interesting than the Dolmabahçe Palace and which requires a bit of a trek to go visit.
If you have the time, I would recommend wandering over to the more rare sights in the Western side of the city such as Chora Church and Pammakaristos Church. My wife and I had a lot of fun just wandering around in that part of the city since it is definitely off the beaten track and is older and less European looking.
I highly recommend a visit to Anadolu Kavağı, a fishing village on the Asian side of the Bosphorus up a short ways from Istanbul. You can take a ferry directly there in the summer but if you’re feeling adventurous try it the way I did it: commuter ferry across the river and two local buses up the coast. When you get there, go up the hill to the abandoned castle. The view from there to where the Bosphorus meets the Black Sea is absolutely breathtaking. And there are hammocks you can rest on on your way down 
Wow, thanks for all the responses!
At this point, I think this deserves a change of fora.
ArchiveGuy, if you take Caffeine.addict’s advice and see Chora Church (which I highly recommend, it is breathtakingly beautiful in a more intimate way than Hagia Sophia, mainly because it’s smaller and there aren’t one gazillion people there with you), there’s a lovely place to eat just next-door called Asitane. It is a bit expensive but it serves authentic Ottoman-era recipes from the kitchen archives at Topkapi palace. I got really bored with Turkish food after spending my five weeks there, but the food at Asitane was much different, and very good.
The Istanbul Archaeology Museum is the best museum I’ve ever been to, especially the Ancient Orient display. I don’t think anything outside the British Museum or the Pergamon Museum could possibly top it. It’s at Topkapi Palace but it’s a separate entry fee to get in. Still totally worth it.
I also really enjoyed the Mosaics museum, near the Blue Mosque. I went to Turkey with a mosaics fanatic and we saw a lot of them while there, but this one had the best!
There’s five or six entries in my travelpod blog from my trip that focus just on Istanbul if you want to have a peek at some of the things mentioned by various posters 
My wife and kids and I are going at the end of July for a week. We are really looking forward to it. I’m glad to hear that English is understood and we’ll be able to get by. Fortunately we will also be accompanied by a Turkish citizen, so we have no concerns.
- Monday to Friday we will be looking at the most of the sights mentioned above in Istanbul
- Saturday we will fly to Izmir and then onto to Kuşadasi to see Ephesus and have a bit of relaxation after Istanbul.
- Monday we will fly to Kayseri and then onto Cappadocia to see the rock formations & underground cities.
- Wednesday morning we fly from Kayseri to Istanbul to connect with our flight to London.
My wife has wanted to visit for years, and the fact that we are able to take our 9 year old twins is fantastic!
BTW, just because they use the Latin alphabet doesn’t mean things are pronouncable. Our hosts name is Mübeccel, which is (roughly) pronounced Mudjella.
I lived in Istanbul for a few months with only minimal Turkish skills, so you should be fine. It’s well worth your while to try to learn a few basic phrases, however!
To ask for something, say “__ isteyorum”. For example: “Su isteyorum” (I want water)
“Pardon” means “pardon me, excuse me.”
Quick and dirty on the pronunciation of Turkish: Turkish uses the Roman alphabet with some modifications. The most obvious ones you need to know are C (pronounced like our letter J), the dotted İ, the undotted I, and Ğ, which the Turks call yumuşak ge and which I call "the g with the little doodad on top of it).
Ğ simply lengthens the vowel sound in front of it. For instance, “sağ olun”, which means “Remain healthy” is pronounced “Saa olun.”
Turkish dotted İ is pronounced “ee”. Turkish also has an undotted I, in lowercase ı, which makes this “uh” sound pronounced a little down in your throat. Don’t worry if you can’t get it perfect, just use an “uh” and they’ll understand you. Watch out: the undotted ı is at the same place on their keyboard that our regular i is, but if you’re trying to log-in to your email and use the undotted ı, it won’t work! Double-check.