You can go to dharamshala and Auli
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Hi Sabera, some of the popular hill stations near Delhi are Bir-Billing, Kasol, Dharamkot and Khajjiar. If you want to go to slightly off-the-beaten-path hill stations then the options are Kausani, Naukuchiatal, Gushaini and Parwanoo. These hill stations have a wide variety of accommodation options from hostels to hotels and AirBnBs to homestays. Following hill stations have pet-friendly accommodation options mentioned in brackets - Shogi (Vatika Resorts and Oakwood Hamlet), Jangaliyagaon (Emerald Trail), Jibhi (Jibhi Cafe & Cottages), Kullu (The Himalayan Village), Gagar (White Peak Homestay), Kanatal (Club Mahindra Resort), and Vanghat in Uttarakhand. You can also refer to the attached article for more hill station options near Delhi.
Yes i had treked with my pet to kheerganga . it was an nice experience
Hi Sabera, I have updated the answer with hill-stations with pet-friendly accommodation options
Hi Sabera, now I don't have a pet
Yes June to oct is the best time to go
Sorry but there is no hill station near New Delhi.