You can go to Nag Tibba which is about 75 kms from Mussoorie. It's a good trek for beginners (9.5 kms). There is a stone trail along with some patches of forest area till the first base camp (7,627 feet) and above that (up to 9,915 feet) the whole terrain is in forest area where the shade of the trees relieves you from the excruciating heat. One can totally experience the best of nature with great flora and might get lucky enough to sight some wild animals (who are by the way completely harmless) Going from the first base camp to the Nag Tibba Summit, there is an old temple which is located about 1 km before the peak. So, it can be a great place according to your requirement.
Near Mussoorie you can go to following places:
1. Kempty Falls
2. Tiger Falls (a must go place)
3. Deoban 4. Chakrata
5. Dhanaulti(A road trip) 6. George Everest House 7. Lambi Dehra Mines(Haunted Place)