From the airport, take the number 100 Airport bus to Princes Street http://www.flybybus.com/.
Bus services 7, 10, 12, 14, 16, 22 and the 25 travel down Leith walk from Edinburgh centre. Ring the bell to get off when you see Tesco Metro on your right hand side (a short distance down leith walk). Then cross the road and face the direction the bus was going, The street is on your right hand side.
Otherwise you can take a taxi from the City centre for approximately £5 - £6 or from the Airport for approximately £20.
From Waverley train station or Edinburgh bus station, make your way to Princes Street to get any of the above buses.