I use this website for finding out easy transport. Most of the time it works for me. https://www.rome2rio.com/ (don't forget to change the currency to INR in settings tab)
Hi Faiz, the cheapest way to travel between from Dehradun to Kasol is to take an HRTC (13 hours, Rs. 700) bus from Dehradun to Kullu. From Kullu, you can either rent a bike and ride to Kasol or take another HRTC bus to Kasol. You can also rent bikes in Kasol. There are several bike rental agencies. The one that I know of is Rahul Thakur (8988265439).
Himachal Tourism (HPRTC) Buses are the most cheapest way of traveling in himachal pradesh. The Drivers are really talented and very good at handling the situation.
Publiy transportation- the cheapest and the best always. You can go to Chandigarh via non-volvo bus and from Chandigarh to bhuntar volvos are available and afterwards you can use public buses or taxi to kasol and manikaran. I also suggest you to go kheerganga and have their hot spring water bath.