Edinburgh Airport: Take the Airlink 100 from stop A to Waverley Bridge via Haymarket and Princes Street: 20–30 minutes, 24 hours, roughly £6 single or £8.50 return. It is quicker and cheaper than the tram for most central hotels. Use the tram only if its stop is materially closer to where you are staying, or if you are headed to Leith; airport-to-centre is about 40 minutes and roughly £7.90 single.
Edinburgh Waverley: This is the central station, below Princes Street and the Old Town. Walk to most central hotels and sights; Waverley Bridge is immediately outside for the Airlink 100 and city buses. Expect steep exits and stairs—use a taxi only if luggage plus an Old Town hill is going to ruin your arrival.
Haymarket: Use Haymarket if your hotel is in the West End, around Shandwick Place, or near Murrayfield. It is one stop west of Waverley on many intercity services and sits directly on the airport tram/Airlink corridor; central Old Town is a 20–30 minute walk or a short tram/bus ride.