Hey Sourabh, First of all, all the very best for your mountaineering course in one of the best institutes in India.... Coming to your question, I wouldn't suggest you to go on a trek with the left over 15 days, as HMI would have taken you through all kind of Trekking practices and rock climbing stuff. Believe me you would need something else at that time ... Go backpacking around places like :- Gangtok Kalimpong Kurseong West Sikkim Mirik South Sikkim Lataguri Jalpaiguri Jaldapara wild life sanctuary Dooars in west bengal In all the above places u can get mesmerizing views of mountain ranges of Himalaya and panorama of breathtaking sun rises all the days ... Visit water falls, monasteries, temples, bust clad streets and savour street foods in all places ... Enjoy backpacking in this part of world and stay in zostels and hostels and also Air Bnbs which would perfectly suit your shoe string budget ... Public transportation is the best mode to rely upon in this part of India ... Happy Travelling Bro !!!
Hey Sourabh, No man ... by the time I was educated about mountains and their majesticity ... I was too old and had no time to take the course ... But have done a couple of Himalayan treks like kuari and rupin passes. ...
Sikkim is very near to Darjeeling ND there are a lot of treks available in the state like the Goecha La, Dzongri, Sandakphu etc. Though you ll have to do a detailed research of the status of these treks for the season. But apart from that Sikkim is a great place to be.