The distance from Dehradun to Mussoorie is 30 kms only and it takes about 1+ hours due to hill road from Dehradun to Mussoorie.
In Mussoorie , one can see Gun Hill, company garden , Lal Tibba and Kempty Falls etc. The shoping can be done on mall road, landaur bazaar, camel road etc area.
Dhanaulti, Chamba and Tehri Lake are nearby areas to Visit from Mussoorie. Dehradun is also good place to see on the way back or while going to Mussoorie for shopping and roaming around.
I hope that will solve your enquiry.