Start your day near Sultanahmet, where many of Istanbul's main sites are located. First check out Topkapi Palace. Spend two to three hours exploring this fabulous Ottoman palace. If possible, try to take in the Harem tour. Afterwards head up the hill to Hagia Sophia and spend an hour there taking in the mozaics.
A short walk from Hagia Sophia is the Bindirdirek Cistern. It's not quite as spectacular as the more popular Basilica Cistern, but you will not have to waste any time waiting in line. Spend an hour here soaking up the unique atmosphere and then grab a quick lunch at one of the many street vendors. After lunch, head to the Blue Mosque along Sultanahmet Square. You can easily spend a couple hours marveling at Istanbul's signature landmark. Before heading across the Golden Horn, take some time to explore the monuments of the old Hippodrome.
Next take a taxi or tram to the historic Galata District. Head to the top of the Galata Tower and take in the marvelous views of Istanbul. When you're finished, head to Istiklal Avenue, the beating heart of the city. Have dinner at one of the many excellent restaurants that line this bustling street. This is also Istanbul's best shopping district, so don't forget to pick up some souvenirs. End your day with a sunset stroll along the Bosphorus.