Istanbul is awesome. It's become quite touristy,though.
1: Hagia Sophia. One of the most impressive buildings on earth, especially considering when it was built!. This was pretty much undeveloped as a tourist site first time I was there some 20+ years ago and I almost had the place to myself!. It's a lot more more busy now, but still good.
2: Topkapi Palace. We couldn't get in last time, due to some official function, but I got a stack of nice pictures from my first trip there.
3: Sultan Ahmed Mosque (Blue Mosque). It's a functioning mosque and as such a bit limited in access as the main hall is always in use.
4: Basilica Cisterns. This is really nice. Roman cistern
5: Galata Tower. Kept missing this, so you should go
6: Princess Island. The ferry ride was nice and there's a nice little orthodox monastery on top of the hill.
7: Grand Bazaar. My wife bought a watch there. It was broken by the time we left and then took us 2 hours to haggle a refund. If you see anything you want to buy, you can most likely buy it cheaper elsewhere, but it's still impressive. Leather market actually looked good, but I was burned out by then. Expect louts of touts, but it's not as bad as say, Marrakesh.
Just walking around the Hippodrome is nice too. I would stay in the old town if you can, but public transit is decent otherwise.
I also always wanted to explore the original city walls.
Drivers seem quite mad, so I wouldn't cycle.
Street food is quite good. Favourite is the guys selling grilled mackerel sandwiches, which are really nice.