I strongly recommend using public transport over driving or taking a taxi. The traffic situation in the city center has become pretty bad in the past decade, and not many good things can be said about Prague taxi drivers. If you're going to be staying in Prague for a few days, you may want to buy a tourist pass that will enable you to use any type of mass transit as many times as you want within the time period for which your pass is valid.
Airport to city. Prague airport is serviced by two buses: no. 119 and no. 100. Taking the bus is the cheapest way to get to and from the airport. The ride costs 20 Kč and both bus stops are easy to find. Bus 119 is the better choice if your destination is in the historical center of Prague (the Old Town, Lesser Town, New Town, Prague Castle area) as it connects with line A of the metro. To take either of the buses, you will need a public transport ticket for 20 Kč (transfer ticket) - or you can pay the bus driver directly, in which case you will be charged a small extra fee. Keep in mind that the bus doesn't have much space for luggage, can be crowded in peak times and won't drop you off right at your hotel, so if you have a lot of luggage, you may want to consider using a transfer service or as a last resort, a taxi.
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Using one of the Prague airport transfer companies is a good option if you'd like a door-to-door service and prefer to arrange for your transfer in advance. Unlike taxi companies, the shuttle companies require you to book your airport transfer in advance. Taking a taxi in Prague is not a simple business. Please read our page, and you need to be very careful to avoid getting ripped off and having a bad experience right after you step off your flight. I strongly discourage you from getting into a random taxi that's parked in front of the airport terminal. Some hotels have a pick-up service.
Around the city. The Prague metro is fast, efficient, clean and easy to use. Its three lines consist of about 50 km of tracks running mostly underground, and some 50 stations. Trains run from 5 a.m. till midnight at 2-3 min. intervals during peak hours and at 4-10 min. intervals in the off hours. There are three transfer stations.
Trams (tramvaje) cover a large area of the city with 500 km of tracks. Daytime trams run from 4:30 a.m. till midnight at 8-10 min. intervals. Night trams (numbers 51-58) run from 00:30 a.m. till 4:30 a.m. at 40 min. intervals. Trams 22 and 23 follow one of Prague's most scenic routes, passing by the National Theater to Staroměstská and Malostranská metro stations, and continuing up to the Belveder, Prague Castle and Pohořelec, with some stunning views on the way. Buses mostly cover the outskirts of Prague.