5 Pieces of Advice on Istanbul

1. The grand bazaar is fun but the odds of you getting a good price are virtually nil. In fact, the duty free shop in the airport has many of the same lamps and tea sets for, in my experience, a third less than the haggled price at the bazaar. In the US, you can purchase some of the same exact lamps at Cost Plus World Market if you'd prefer to skip the worry of getting it home. 
2. The boats that go from the mouth of the Golden Horn are very inexpensive and will take you away from your fellow tourists. Kadaköy offers a more authentic market area without aggressive sales people asking you, “Where are you from?” every 5 steps. The boats from here are part of the transit system and the cheapest way to get form Europe to Asia on the other side.
3. Be sure to check which days and when things are closed! Ayasofia stops admissions at 4pm, and is closed by 5pm. The Basilica Cistern is open later than Ayasofia and is right next to it. The Grand Bazaar is closed on Sundays. The Blue Mosque is an active house of worship, which means that it closes for prayer three times a day. Also, with Ayasofia, it is possible to skip the line if you pay one of the many tour guides that will approach you, but whatever they quote you is in addition to your ticket (the usual fee seems to be 5 to 10TL). Alternatively, there are always large guided tours going on and it would not be hard to simply shadow them to get the same information
4. Taksim square, north of the Golden Horn, is a center for the hip nightlife and packed full of places to eat, live music, get a drink, and more. The most interesting street starts in the southwest of the square. There's a historic streetcar that runs along this street and any of the side streets from here will be good bets for finding restaurants and clubs. 
5. Eating can be fairly inexpensive and a real treat. A baklava break at Karaköy güllüoğlu will cost under 10TL for an assortment of baklava and a tea. It is still the best baklava I've ever had. Döner is also very tasty and ranges between 3 and 7TL. Fresh fish is grilled up by the shore in some parts and can be had as a sandwich as an alternative to the usual chicken or meat döner. Pide is also a treat and a sort of Turkish pizza/taco, usually with a handmade flat bread and then meat and cilantro. Sesame bagels can be had at virtually any square for less than 1TL.