Gujarat will be fun at this time. You can also go to many places in Madhya Pradesh and Maharashtra. I would suggest to avoid mountains and beaches during peak monsoon.
Chopta Tungnath is best place to be planned during this time period. It is situated in Uttarakhand and approx 600 kms from Delhi.
Chandrashila is the main attraction, don't miss it. Amazing place overall and very cheap accommodation.
On a second thought you can go to valley of flowers which is mesmerizing in itself.
Hope this helps.