Visible from almost every part of the city, St. Vitus Cathedral sits within the walls of Prague Castle, the largest ancient castle in the world.
As a Britton and with the styles of castle that I'm used to back home, Prazsky Hrad didn't really have the feel of a castle. In a way, it's more like a palace. There has been a fortified settlement on the site since the 9th century and it was traditionally the official residence of the country’s rulers. Each ruler added to the building, which has led to it comprising a rather odd mixture of styles, including Gothic, Baroque, and Renaissance. It is a magnificent and imposing structure, however you may look at it.
On the hour, you can witness the changing of the guard at the front gates. This is most impressive at noon because there is music played by brass musicians and all the guards march down the street and into the courtyard where banners are exchanged. The views over the city from the road leading to the castle gates are excellent and there is usually some sort of musical ensemble playing to entertain the crowds waiting to see the changing of the guard. There are three different types of ticket available, with each allowing entry to different areas of the castle and cathedral. Ticket A allows access to pretty much everything and costs 220kc (£4.90/$7.30) or half that for concessions.
The most obvious of the castle’s attractions is St. Vitus Cathedral. It's possible to climb the Great Tower, but I have to say that I didn't find this a very enjoyable experience. The dark staircase is very narrow and claustrophobic and the views from the top are no better than those from other viewpoints in the city. However, the cathedral is very impressive and the colourful stained glass windows are beautiful. Take about half a day to enjoy the cathedral and the different areas of the castle such as The Old Palace, St. George Square, the Basilica of St. George, Golden Lane, the Powder Tower, and the Royal Garden.
One way to reach the castle from other areas of the city and make use of public transport is to take the metro to Malostranska and then take tram 22, 23 or 57 to Malostranske namesti. The walk along Nerudova to the castle is nice (although fairly steep) and takes you past many embassies, Czech pubs and restaurants, and interesting shops.