Turks love children and fawn all over them as a matter of course. Taken to the not-so-extreme, a total stranger may stroke your child's hair to show his or her benign affection. Back home, this would set off alarm bells; in Turkey, this is actually just a cultural norm. As for sightseeing, Istanbul is a veritable treasure chest of the magical and exotic, making exploration with the brood a delight for all involved. Kids seem to love scrambling around toppled columns and ancient ruins, exploring underground cisterns (especially the ones with water in them), and poking around the very colorful and stimulating Grand Bazaar.
If you're planning on renting a car or getting around with a tour company's car, keep in mind that car seats are considered a luxury item in Turkey, so ask/reserve ahead of time to be sure you can get one. Almost all hotels offer cots on request to accommodate families together in a room, and many restaurants have high chairs (when they don't they'll find some way to improvise). Other baby and toddler items are widely available (diapers, UHT milk), but if you're set on using a particular brand, better bring it along. Bottled baby food can be difficult to find, but restaurants will probably be more than willing to puree whatever you want.
Meanwhile, almost all tourist services (hotels, ferryboats) offer some kind of discount for children, usually 50% off the full price for children 6 to 12, while kids under 6 generally get loads of freebies. Unfortunately, this discount does not apply to admission tickets to museums and sites for preteens. You'll also save a load of dough thanks to the Turkish tradition of providing breakfast with the room.