If you really want to live on budget and spend awesome 3 months in mountains away from all the crowd and hustle bustle of city, try Thirthan (twin valley for more famed parvati). You will enjoy every bit of it. Its not much explored by tourists (people who really like to travel not for social media update)
Hi Sindhu, I would recommend Kasol, Himachal Pradesh for your 3 months long budget trip. You can get the hotels/rooms as low as 300-400 bucks a day. You can access all the places by traveling on local buses. Most importantly, wherever you must spend your time at a place where you get n number of options to visit and Kasol is most suitable for it.
If you want to match cost of living and a hill station in South India Kodaikanal is best, as rentals are low, you will get freshly cultivated veggies plus a beautiful lake perfect for an evening stroll and a morning walk. The weather is not soo cold and remain pleasant through out the year. Location wise also it's around 3 hour journey from Madurai Airport and also there are daily buses to Chennai as well as Bangalore. if it is north there are plenty of choices in Himachal and Uttarakhand.
Do come to mukteshwar. .. it is a wonderful place to spend time . you will love the ambience and pristine nature.. campsites... mesmerising view of himalayas. you can even visit nearby places like sitla.. Ramgarh... kausani... someshwar... hippy hill ... jageshwar... baijnath.