Day 1 - 5 Istanbul
Staying in the Sultan Ahmet area (good choice) and will take in the following sites that we can walk to from our hotel:
Topkapi (where can I get tickets to get into the Harem part and do I have to book this part through a tour? What tour group do you recommend?)
Start early for the tour of Topkapi palace. The harem tour can be booked at the entrance part, when you pay for the general entrance ticket. I assume you are English speaking so book an English speaking Harem tour here as well. You can walk through the rest of Topkapi at your own pace, with the help of audio guides that are also available at the entrance. The restaurant at the far end of the palace is a scenic spot to rest and have a food and drink break. Allow at least 1/2 a day to see this sight.
Hagia Sofia can be seen self guided. Make sure you go up the ramp to the upper galleries to see the excellent mosaics up there. Allow 1/4 day (approx. 2 hours)
When you leave, a very short distance away when you cross the street to your right is the Basilica Cistern. In my opinion this is also well worth experiencing. The entrance is a small kiosk like building on Yerebatan cad. (street). Allow roughly 1 hour to see this very interesting ex water storage space.
Grand Bazaar : A huge complex maze of shops. Experience it the first time without haggling for ´bargains´. Save that for your return later in your trip when you might be more experienced and more aware of good prices. Don´t get trapped in carpet shops with salesmen. All salesmen are well versed in engaging tourists and you will waste a lot of time when you are friendly enough to talk with them. Allow 2-3 hours, depending if you are only browsing, not buying.
Blue Mosque bring your own headscarf (for females) if possible, and be appropriately dressed. Avoid the prayer times (recognised by prayer calls!), and a free guide will often explain the mosque to you. A donation for mosque upkeep costs can be made once back outside the mosque when you put your shoes back on. (Allow roughly 1 hour. See it early to avoid the long queues when tour groups arrive. If the queue is too long, don´t waste your time, come back later, especially as you are staying in the area.
Sulaymanye Mosque (near Grand bazaar) Again avoid prayer times.
Possibly:
Dolmabahce Palace: definitely worth touring for its magnificence. A slightly different (newer) feel than Topkapi Palace. Allow at least 1/2 day.
Spice Bazaar (I´m not really interested in spices so I think we might skip this one) In my opinion don´t skip it, as you have several days in Istanbul. Bring your camera. It is more about the feast of colour for your eyes! It sells food and sweets as well as spices, and feels more like a continuation of the covered bazaar. Combine this with a walk around the Galata bridge. Allow 1 hour.
Princes Island (If we have time, I think I can book a tour when I get to the hotel) . Will take most of the day. Go early. Just grab the fast ferry from Kabataþ (near Dolmabahçe Palace). Hop off the ferry at Büyük Ada. Horse and carriage tours around the island can be taken near the village square. Just follow the other tourists. Allow 3/4 day minimum, by the time you return. From the older historic ferry wharf, take the slower ferry via Kadýköy to Eminönü to get back closer to your hotel.
Taksim Square (not sure what is there?) A large monument in the centre of a roundabout, but this is a good spot to start to tour down Istiklal cad. I think you can catch the train from Sultanahmet down the hill and across the Galata bridge to near Kabataþ ferry wharf. Hop off and there is a underground station where you can catch a "new" 1 stop underground train to the top of Taksim Hill. Walk across the square and to your left you will probably see a tram. Then walk down the hill via Istiklal cad. Follow the Tunel tram lines. Lots of modern shops, bookstores and food places. Excellent restaurants off the side streets. Where the tram stops is an entrance to an old funicular railway which will take you to the bottom of the hill near the Galata bridge. Don´t forget to visit the open air Fish markets on your side of the bridge. Lots of gun shops and electronic goods if you walk the lower arcades of the Galata bridge from the fish markets. Walk back across the bridge to Sutanahmet and experience the loads of people fishing from the upper deck.
Hope this helps you, and I know you will enjoy your experience in Istanbul.
Turkish people are very friendly and are generally always helpful.