Hi..... nupur yes definitely you can travel from nanital to dehradun, mussoorie, Rishikesh, etc. by public transport. frequent buses available from nainital, haldwani to these above mentioned places. even you can go through railways. some travels you can catch from kathgodam (nainital) to dehradun, haridwar etc. dehradun is based camp for mussoorie kathgodam (haldwani) is base camp for nanital. connectivity not an issue here. go for it 👍
Nainital to Rishikesh is 9 hrs travel there are buses available which you can check via Yatra and goibibo and ixigo too. From Rishikesh to Mussoorie best way is taking a cab for its 1 hour drive. From Mussoorie Dehradun is 2 hours away. Next to Dehradun railway station there is a stop where buses come for Dehradun. Choose the route Yamunotri and bagkot to reach Dehradun. Else book a cab from Mussoorie to reach Dehradun.