With only one day to explore İstanbul you will only be able to focus on one or two major attractions. 

The most obvious bet would be Topkapı Palace where you could easily spend most of a day. If you have any energy left afterwards you could also pop into the Blue Mosque and take a turn around the historic Hippodrome or Gülhane Park, places without opening hours to worry about. 

If you've already seen the palace and the weather is good you could instead spend a happy day one of the long Bosphorus boat rides that take you as far as Anadolu Kavağı whence you can climb the hill and gaze out over the Black Sea.

Alternatively, you could explore the other historic sites of Sultanahmet - Hagia Sophia, the Blue Mosque, the Yerebatan Cistern, the Archaeological Museum and the Great Palace Mosaic Museum - as they are all handily close together. If that is your plan you could even drop into the Ayasofya Hürrem Sultan Hamamı right in Sultanahmet Square to try out a Turkish bath originally designed for the wife of Sultan Süleyman the Magnificent. Don't expect it be cheap though. 

Shopping

If you want to devote your day to shopping there are several obvious options. It would be easy, if exhausting, to spend the whole day in the Kapalı Çarşı (Grand Bazaar) although that's a rather touristy experience. 

More managable are the Mısr Çarşısı (Egyptian or Spice Market) at Eminönu or the Arasta Bazaar beside the Blue Mosque. If you visit either of these you will still have at least half a day left for other sightseeing.

More interesting would be to take the tram and funicular to Tünel where you can explore the many small shops and trendy boutiques around the Galata Tower. 

Eating

If time is tight and you don't know the city well you might want to eat in Sultanahmet rather than venturing further afield and worrying about transport connections.

For something fairly authentic and not too expensive try Tarihi Sultanahmet Köftecisi Selim Usta on Divan Yolu which has been dishing up meatballs to enthusiastic locals since 1920.

Alternatively in summer you could take a taxi to nearby Kumkapı or Samatya to eat outdoors at one of the many fish restaurants.

Three days is enough to explore İstanbul's main highlights. This is what I would suggest:

DAY ONE

Spend most of the day in leisurely exploration of the Topkapı Palace including the Harem and Treasury. Afterwards you can pop into the Blue Mosque and take a turn around the historic Hippodrome or Gülhane Park.

DAY TWO

Explore the other historic sites of Sultanahmet - Hagia Sophia, the Blue Mosque, the Yerebatan Cistern, the Archaeological Museum and the Great Palace Mosaic Museum - which are all handily close together. You could also drop into the Ayasofya Hürrem Sultan Hamamı right in Sultanahmet Square to try out a Turkish bath originally designed for the wife of Sultan Süleyman the Magnificent. If it's too expensive walk along Divan Yolu to Çemberlitaş where another 16th-century Turkish bath designed by Sinan is cheaper (although not cheap).

DAY THREE

If the weather is good you can spend a happy day one of the long Bosphorus boat rides that take you as far as Anadolu Kavağı whence you can climb the hill and gaze out over the Black Sea.

If the weather fails you can travel out to Edirnekapı to visit the Chora Church (closed Wednesdays) with its magnificent Byzantine mosaics and murals. Nearby you will be able to explore a stretch of the Theodosian City Walls and perhaps visit the new Panorama 1453 Museum to get an eye of what it was like on the way in 1453 when the walls were finally breached by the Ottoman troops led by Mehmed the Conqueror. 

This should leave you enough time to travel across to Taksim Square (take the tram from Sultanahmet to Kabataş and then the funicular to Taksim) and walk along pedestrianised İstiklal Caddesi which is always heaving with people and which is lined with shops and restaurants offering something for all tastes and budgets. It's a great introduction to modern Turkey and an antidote to all the tourism of Sultanahmet. 

Shopping

With three days up your sleeve you will have greater flexibility when it comes to shopping. Some people will be itching to get into the Kapalı Çarşı (Grand Bazaar) although it's a rather touristy experience. 

More managable are the Mısr Çarşısı (Egyptian or Spice Market) at Eminönu or the Arasta Bazaar beside the Blue Mosque. If you visit either of these you will still have at least half a day left for other sightseeing.

More interesting would be to take the tram and funicular to Tünel where you can explore the many small shops and trendy boutiques around the Galata Tower. 

Eating

If time is tight and you don't know the city well you might want to eat in Sultanahmet rather than venturing further afield and worrying about transport connections.

For something fairly authentic and not too expensive try Tarihi Sultanahmet Köftecisi Selim Usta on Divan Yolu which has been dishing up meatballs to enthusiastic locals since 1920.

Alternatively in summer you could take a taxi to nearby Kumkapı or Samatya to eat outdoors at one of the many fish restaurants.