Daily direct Train runs from New Delhi to Jammu and few traing runs directly to Shri Vaishno Devi (Katra). Please check IRCTC.CO.IN for train schedule.
There are many buses plying to Jammu from various parts of the North India. From Jammu you can catch a local bus or hire a Taxi to Katra.
The nearest airport is Jammu and daily many flights are from Delhi to Jammu. From Jammu you can catch a local bus or hire a Taxi to Katra.