Which will be the cheapest way to go to Leh from Jaipur and how? It's my first time to go on such trip. I am so confused.
Bus to Manali via Keylong, two days (overnight in Keylong), Rs 700 or so on the public HRTC buses, around Rs 2000 on the plusher private ones. Bus from Manali to Delhi, ~14 hours, Rs 500 up. And finally train to Jaipur, 4+ hours, fares from Rs 150 or so (more on the expresses).
Alternatively, bypass the dangerous twisty mountain roads and fly to Delhi, then take the train. Fares vary, but will be of the same order of magnitude as taking nice private buses, and far faster and more comfortable.