You could take a bus to Rishikesh and then from there to Mussoorie transport is available. Another way is to Haridwar same way as above. There are tour operators in the town who might help you with road travel. You can book via Goibibo or Doon cabs or Savaari rentals. There is a shatabdi and Mussoorie express leaving from Dehradun to Mussoorie. Kindly check the timings.
You can find busses right on the highway near the airport which go to various destinations. Most of them to Rishikesh. It's very frequently available, so you may hop on to any of these buses. They would stop for you.
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