Edinburgh has it's own airport (EDI). Other airports of interest might be Glasgow, and Prestwick.
A good website to use for finding flights is http://www.skyscanner.net (Rob used to work for them, so of course it's good)
There is a frequent shuttle bus service from the airport to the city centre. http://www.flybybus.com/
.. and if you lose your luggage, lost luggage people around here are really committed.
The main train station in Edinburgh is Waverley Train Station.
A good way to find your public transport connection is either Google maps, or http://www.travelinescotland.com/
The main bus service in Edinburgh is run by Lothian Buses. They are good for getting around Edinburgh, but it can get a bit confusing for visitors to work out the bus routes.. but bus drivers are normally really helpful and patient. Here are our tips:
Edinburgh is full of black cabs, which can be flagged on the street (if the orange 'Taxi' light is on, it means it's available). A good company is City Cabs which can be called on 0131 228 1211.