Sorry Im a little late joining this thread but youve been given excellent advice so far.
Sultanahmet is certainly the place to stay as you are within easy walking distance of the citys main attractions. I can recommend a hotel right in the heart of Siultahmet where we have stayed many time and members of our family and friends. Have a look at their website here.
They have Hotel and Pansiyon rooms and offer comfortable and spotlessly clean accommodation in a secure environment with customer-friendly staff and offer a great deal of helpful ,local information. It is a few metres from the Four Seasons Hotel at a fraction of the price and extremely close to:Aya Sofia
: One of the most impressive and important buildings ever constructed. For almost a thousand years, Aya Sofia was the largest enclosed space in the world. The building has been a mosque and a church over time. Aya Sofya is impressive one of the first things most visitors want to see when they're in Istanbul.Topkapı Palace:
Home of the Ottoman Sultans for nearly 400 years, Topkapı Sarayı was the heart of the vast Ottoman Empire, ruled by the monarchs who lived in Topkapı's hundreds of rooms with hundreds of concubines, children, and white and black servants. Don't miss the Treasury with its incredible gems, gold, and works of art.Blue Mosque:
Free entrance, but donations will be requested. With six minarets it was said to rival the design of the mosque at Mecca and the mosques imposing size is impressive. Basilica Underground Cistern.
Ancient water storage with over 21 million gallons of water and 336 marble columns. In the James Bond film From Russia with Love, the scene when Bond is rowing in a small boat through a forest of marble columns was filmed here Grand Bazaar.
One of the largest covered shopping markets in the world with 60 streets and 5,000 shops. Free (unless you buy anything, of course!)
As you are only visiting for a few days ou may also want to consider the Short Circle
Bosphoros Cruise, along the straits of water separating Europe and Asia towards the Black Sea and lasting about two hours. You'll see Galata Tower, Topkapi Palace, Maiden's Tower (also seen in "From Russia with Love"), the Selimiye Barracks (where Florence Nightingale worked), Dolmabahçe Palace, the Bosphorus Bridge (linking Europe and Asia) and much, much more. The ferry leaves the Eminönü docks at 14:30, arrives at Üsküdar around 14:50 to pick up passengers from the Asian side, and then continues its two hour non-stop tour.
Do let me know if I can help with any more detailed information. I am sure you will have a great time in this fascinating city.