June is summer break for Schools in Northern India. Expect lost of rush at this time and its hard to find a decent place for stay if you don't book in advance. Last time Shimla went out of water during the summer break. I suggest you to look at other options in Himachal like Khajjiar, Dharmashala, Mcleodganj.
For adventure sports such as paragliding/rafting this would the best time!
However if you want to witness the beautiful valley covered in snow blankets I would suggest you should plan it in month of December/January.
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