You can visit any of the himalayan states. If you're visiting during the first two weeks, you can visit any places in Himachal, Sikkim, Uttarakhand or Jammu and Kashmir. Try avoiding Sikkim and parts of Himachal on the last weeks of June as heavy rains start due to monsoon.
I think that depends on where will you be travelling from and what's your budget? Northern and Southern India offer a plethora of options. While in North you can always go for Kufri, a little further up Shimla. It's known for adventure sports. But I would personally suggest Arunachal Pradesh in North East. It's still untouched and hence very surprising. If you want to travel South, there are many small towns which are not as elevated as the Himalayas, but are really pleasant during the summers. Like Coonoor, 20 kms from Ooty. As pretty as Ooty, if not more, and is less crowded than Ooty. You may try Munnar in Kerala. The steps of it's hills grow tea and they are dotted with small churches and huts. The clouds s rise up to the hills making it misty and enchanting. And of course you may try Kodaikanal in Tamil Nadu. Many things remain unexplored here. Like the Dolmen circle, where monoliths stand in mystery. These are 1000s of years old and their purpose or the way they were built are yet not known. I hope these suggestions help. None of these trips are expensive. You can always find a homestay and travel by train or bus Have a happy trip.