The taxi rank is directly outside the airport's main entrance. The journey to King's Buildings takes about 20–30 minutes and will cost around £15.
Lothian Buses operate a frequent shuttle bus (Airlink) between the Airport and the City Centre. During the day, the buses run at least every 10 minutes and the journey takes around 25 minutes. More detailed information on frequency and prices can be found at the Airlink website. Once you get to the city centre, depart the bus when it terminates at Waverley Bridge, just outside Waverley Station.
From there, you can follow the instructions for getting to JCMB by bus from the city centre.