There is no direct train between Bangalore and Kashmir, So passengers have to change train at New Delhi then take any train till Katra, then By Vande Bharat Express or Demu Train one can reach easily
The following steps to reach Kashmir directly
Depending upon different route and different class From Bangalore to Delhi, by premium train Rajdhani or Duronto it will cost Rs 4000–7000 whereas Sleeper Class is approx Rs 380–400
Then Delhi to Katra by Vande Bharat Fare will be Rs 1600–3000, whereas Sleeper fare is Rs 415
Then Katra to Srinagar, Vande Bharat Express fare is Rs 720–1300,
Or best option is Alight at Udhampur then board any bus towards Srinagar/Baramulla, for budget friendly option because From Katra to Srinagar train service local train is less.
If you don't want to spend 2–3 days in train, then travel by flight till Delhi, then take Direct bus to Kashmir Srinagar, it will take 15–16 hours
Flight fare from Bangalore to Delhi (non stop service) will be Rs 8000–9000.
If you are budget constraint and ok to manage timing, then Sleeper Train option till Katra (By changing Trains at Delhi) would be good option,
I don't recommend you to travel in unreserved coach in such a long distance journey.