The Castle is one of the most famous and most historic images of the city. It attracts all kind of people, and especially visitors. Locals love there because it has a great view through Youth Park (Gençlik Parkı), which is another important place of Ankara, and through the city. Also, you can see the traditional houses there.
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You can take some photos, have a traditional breakfast or lunch, watch the sunset while looking through the city. While visiting a historical place, you can also see the cultural side of that part, and the economic distinction between social classes. Below the Castle, there are numerous souvenirs, if you want to keep something as a memory.
You can use the public transportation up to a point, but after you'll need a local people to get there. Also the region is not quite safe, be careful.