Good Morning, or as is said in Turkey, "Gun Aydin." How are you?
Turkey is a fantastic country and Istanbul an amazing city. I lived In Istanbul for a year and although I have heard the city has changed in some ways since I was last there (including the addition of a subway system), certain attractions remain timeless. Keep in mind that Istanbul sits on three land masses, two in Europe, one in Asia. You will most likely see mostly the sections of the city that are located in Europe, although a ferry ride across the Bosphorus, the body of water that separates European and Asian Turkey, is a visual treat all its own, given the dramatic views of the three great mosques in the "Old City" and Topkapi from the water. You may want to do this, just for the view. Go at sunset.
Here is my short list:
Topkapi, the Sultan's Palace
Ayasofia (St. Sophia)
The Blue Mosque
The Grand, Covered Bazaar
The Walls of Old Constantinople
Dolmabace, the other Sultan's Palace
Kariye (Chora) Church
The Archeology Museum
There are also islands off the Sea of Marmara on which many "Istanbulus" (as they say in Turkish) maintain summer homes. all of which are accessible by ferry from Istanbul. The most famous, Buyukada (Big Island) might be worth considering for a weekend day trip. There are no cars, generally only bicycles and horse drawn carriages.
As for places to stay, Taksim is a good choice although you will not be within walking distance of old Istanbul and will need to grab a taxi or bus it to many of the big sites. Beyolgu is another central, trendy neighborhood that you should look at. For seafood, try the Kumkapi area, which is where one can find some good fish restaurants. If tconvenience is important to you, you should check out hotels in Sirkeci, which is the neighborhood close to Topkapi Palace. If good shopping is key, then try NIsatasi (pronounced Nishantash).
And when you need a break from all that walking, find a coffeehouse, have a seat, grab a Turkish coffee or their ubiqitous "chay" (tea), play a game of backgammon or chess and relax for a bit.
This is a short list but I hope you find it useful. If you have questions, feel free to hit me up.