Dalkeith is a town in Midlothian in the Central Belt of Scotland, which grew up near its 12th C castle (which was replaced by the 18th C Dalkeith Palace). It's nowadays a commuter town for Edinburgh, which is six miles northwest. This was a coal mining area in the 19th C and a couple of miles south in Newtongrange is Scotland's National Mining Museum.
From the airport, follow ring-road A720 to congested Sheriffhall roundabout then A6106 for the last mile into Dalkeith. You can also take Skylink Bus 400 to Fort Kinnaird on the southern edge of the city, then change to the buses below.