I prefer to fly out of DXB Terminal 1 or 3 whenever possible because of the metro line, with trains departing often from 6am to midnight. For Terminal 2 its either bus travel or a metro ride near the airport to a bus/taxi; thankfully taxis are not very expensive in Dubai, but traffic can raise the fares considerably. For some reason the city cancelled the only bus route from main Ghubaiba bus station to Terminal 2, but other routes do exist; check a current schedule as times and routes often change here. My advice is give yourself plenty of extra time if you are planning to get to the airport (or anywhere else) by bus.
The new airport is Dubai World Central (DWC) -Al Maktoum International Airport which is around 50km west of central Dubai near the Jebel Ali area. Flights just started in October 2013 and there is an interesting mix of airlines flying out of DWC, including the Eastern European budget airline Wizz Air. The only public transportation option to this airport is Bus F55 which originates in front of the IBN Battuta metro station (the other bus stops are not very convenient unless you happen to live near one of them). If going by metro to bus to DWC keep in mind the travel times! The metro ride from Dubia's city center to IBN Battuta takes about 1 hour and then you have the bus connection. The F55 bus schedule says it takes 45 minutes to get to DWC from the stop out front of the metro station and there is only 1 bus per hour, so once again leave plenty of time! A taxi from IBN Battuta or Jebel Ali metro stations is another option. A taxi from the city center has to contend with the traffic and toll road system so is best avoided in my opinion.
Side note: At writing Wizz Air is offering flights from DWC to/from Budapest, Bucharest, Sofia or Kiev for a little over 100 euros each way! Keep on traveling!
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