There are lots of must visit places in Istanbul. I think you should not spend less than 4 days there. But if you have only 2 full days, then here is my suggestion based on their vicinity.
– The Hippodrome of Constantinople (Sultanahmet Meydanı) which was a circus that was the sporting and social centre of Constantinople, capital of the Byzantine Empire. It public space, opens 24/7, perfect to start your day. No entrance ticket.
– Topkapi Palace = great place to learn history of Ottoman Empire, the last khilafah which governed three most holy cities for Muslim i.e. (1) Makkah, (2) Madinah, (3) Al Quds (Yerussalem). Unlikely will ever happen again since Israel got their hands on the 3rd city. Time required = 3-4 hours in the morning, recommended to start your visit since the opening hours (9:00 AM). Entrance ticket = 25 TL, additional 15 TL to see the harem too.
– Archaeological Museum (http://www.istanbularkeoloji.gov.tr/main_page). I didn’t visit it because my son wasn’t interested, but seems like you’ll have to be there at least 2 hours. Entrance ticket = 10 TL.
– Hagia Sophia = amazing building which was a church before become a mosque then become museum. Wow effect when I saw the ceilings because Islamic calligraphy didn’t cover places where holy figures in Christianity were painted. Time required = 1 hour. Entrance ticket = 25 TL.
– Blue Mosque or Sultanahmet Cami = really recommended to do your pray here. Time required = 1 hour until Adzan Maghrib to enjoy the sight of the beautiful colorful fountain between the mosque and Hagia Sophia (they didn’t lit the light during the day). No entrance ticket.
Tips: Get “Museum Pass” = 85 TL to visit Chora Museum, Hagia Sophia Museum, Topkapı Palace Museum, İstanbul Archaeological Museums, İstanbul Mosaic Museum, Museum of Turkish and İslamic Arts, Galata Mevlevi House Museum, Yıldız Palace Museum, Museum for the History of Science and Technology in Islam, within 72 hours(http://www.muze.gov.tr/museum_pass).
– Start your day by going to Eyup Sultan Mosque (Bus 99A from Eminonu) early in the morning. Some people believe that when you pray there to ask for something then God will grant it. Time required = 2-3 hours including transport. No entrance ticket.
– Back to Eminonu, cross the street to visit Yeni Cami, then enter and stroll along Spice Market (perfect place to buy Turkish delicacy on cheaper place than anywhere else), end it at Rustempasa Cami (small mosque with beautiful interior). Time required = 2-3 hours. No entrance ticket.
– From Eminonu, ride tram to Kabatas. Get off and walk few minutes to Dolmabahce Palace to visit the residence of last sultan also the room where The Father of Turkiye (Mustafa Kemal Ataturk) passed away. Entrance fee for adult:
Palace only = 30 TL, harem only = 20 TL, palace and harem (combined) = 40 TL. These fares already includes guided tour because visitors were not allowed to walk inside the building by themselves. Time required = 3-4 hours. Don’t forget to make a pose at side of the palace with Bosphorus Strait as your background.
– Back to Kabatas, ride funicular to the famous historical Taksim Square. The nostaljic tram would be interesting experience. It’s a very hype location where you can shop for international branded fashion items and hang out at night clubs.
Other recommended places and things to do if you have extra days:
– Cevahir Mall…Europe’s biggest shopping mall before they opened a bigger one in London (http://traveltips.usatoday.com/top-three-biggest-shopping-malls-europe-104052.html). Accessible by Metro to Sisli. Lots of “indimir” (discount/sale) outlets. Many choice of food/beverage on budget at the foodcourt (5th floor). Perfect for us girls. :p
– Military Museum at Harbiye, Sisli (can be reached from Eminonu by Bus 54E or others, http://placesinistanbul.com/placeTransit.php?placeID=2321&lan=EN) = best place to learn history of Selcuk Empire, Ottoman Empire, modern Turkish Republik. Go there after lunch to see Mehter Band performance which starts at 3:00 PM. Entrance ticket = 5 TL.
– Grand Bazaar near Blue Mosque. No entrance ticket.
– Basilica Cistern, also near Blue Mosque. Entrance ticket = 10 TL. Visitors could make a pose like Sultan/Sultana with provided prop and backdrop (14 TL for one print, 55 TL for one print and a CD contains all of your pics) inside the cistern.
– Bosphorus Tour from Eminonu = 10 TL for 2.5 hours, 25 TL for 6 hours (http://sehirhatlari.com.tr/en/announcement/about-short-bosphorus-tour-929.html).
– Walk cross the Galata Bridge from Eminonu to Galata Tower. I didn’t go there because my 9 years old boy didn’t want to do it, but would be very interesting experience (http://aglobalaffair.com/2012/04/13/a-walk-across-galata-bridge-an-istanbul-must-do/).
– Suleymaniye Cami and the tomb of of Sinan Mimar, also Valens Aqueduct nearby, in Fatih area. I didn’t go there because there was not enough time.
– Take cable car from Eyup Sultan Mosque to Pierre Loti Cafe (http://www.istanbul-ulasim.com.tr/our-network/eyup-piyerloti-aerial-cable-car.aspx). I didn’t go there also because there was not enough time.
– Perhaps visit Istanbul Aquarium, the spectacular tourist attraction (http://www.wittistanbul.com/magazine/istanbul-aquarium-a-spectacular-tourist-attraction/)? I didn’t because failed to get the bus from Eminonu.
– Turkish traditional bath at any hamam (ask your Turkish friend for recommendation)…I didn’t because I had no budget, around 50 TL per person.
– Watch sema ceremony or whirling dervishes (ask your Turkish friend for recommendation)…I didn’t also because I had no budget for the luxury, around 50 TL per person.
I GOTTA BACK TO ISTANBUL, SOMEDAY!