Before you visit, make sure you have a high enough credit limit. The Palace is the most expensive site we've ever visited, anywhere. Admission to all three sections of it (main entrance, Harem, and Treasury) will cost you about $30. Of course, we paid. I understand gouging the tourists (and applaud their understanding of capitalism for doing so - who is going to come to Istanbul and *not* see the Topkapi palace?) but what was unfair was that the prices for students and for Turks was not substantially lower - putting an amazing part of history out of the reach of a lot of people.
Ok, so it's time for a note of warning. The Palace is big. Really, really big. There is a lot to see. And it's all very impressive. Be prepared to spend a full day here, and be well rested beforehand. Bring food and water, as there aren't that many options inside the palace. And bring a lot of film, although they won't let you take pictures in many places.
There are four courtyards to the palace. Only the first is accesible without a ticket, and there is really nothing to see here. The second court contains the Inner Treasury, with gold leaf ceilings and a peep-hole the Sultan once used to spy on arriving VIPs. Off this court you will also find the Imperial Kitchens, and a collection of silver and European procelain. Some of these rooms might be closed; it depends on when you go.
The third court contains the Library of Ahmet III (closed), the School fo the Expeditionary Pages (costumes and kaftans, and the first baggy trousers - isn't it amazing how style comes full circle!) As well, you will find the entrance to the Palace Treasury, and the Hall of the Treasury, with 37 portraits of 37 sultans. If you're not tired and moving at about 10 sultans a minute by now, you're doing better then we did. The treasure dormitory houses Islamic works of art, although it was closed when we were there. Or maybe we'd just had enough. The Pavillion of Holy Relics was probably beautiful, but we were really tired by then. I remember the soundproof glass cubicle from which the muezzin made the call to prayer.
The fourth courtyard has a beautiful Tulip Garden (or whatever happens ot be blooming when you're there.) There is another Iznik (blue) tiled pavillion, the Circumcision Room (where young princes would be snipped), and more upliftingly, a beuatiful view of the Bosphorous.