5 places
The fee is 25 Turkish Lira for entry and well worth it. There are many tour guides that will offer you tours for around another 50 Lira, which is worth it only if you are in a big group. The alternative is to purchase the audio tour, which costs only 15 Lira and can be shared amongst friends.
This is a must-see in Istanbul. The beauty of the building, both the interior and exterior are overwhelming. Not as crowded as the Blue Mosque.
Buy your tickets online in advance (for Topkapi Sarayi as well as the Harem). They are valid on any date or time and you do not have to queue for buying the tickets!
The store owners are all very nice, but be warned that the initial price they give you is not the final price and, in most cases, double what you actually will end up having to pay. Also don't just buy something at the first shop you see it, since most products are available in more than one shop.
Some of the areas are a little slippery, so be careful. There is a nice coffee shop in the bottom in case you need refreshment.