Since it is a bit hot in the month of May in most of the India, You should try going to the hilly areas of north or north eastern India.
Now you have 2 options in this, either you can go to famous hill stations like Shimla, Manali, Darjeeling, Rishikesh etc. where you will feel a good crowd.
or you can choose adventure trips to less crowded cities like Spiti, leh, Bhaderwah (in Jammu) etc. if you are interested in treks, there are a lot of them in the state of Himachal Pradesh like 1. Triund trek
2. Kheerganga trek
3. Parvati valley trek
The hills man the hills and when you talk of mountains it is usually the Himalayas but depending on where you are located, Nilgiris is an excellent option too. However, it is the Himalayas, right from Leh, to all those juicy tourist Hill stations in green Himachal like the famous Manali, Kasol, Parvati valley to the Dharamsala, Dalhousie, to of course Shimla but it would be crowded in May by the local tourist, to the desert terrains of Spiti and Lahaul, in Himachal. Then of course you've Kumaon regions of Uttarakhand apart from Garhwal. In Uttarakhand, one tourist route is to go to the Corbett park and from there continue to Ranikhet which not only boasts of amazing flower gardens but also they highest Hold feeling in the world and from there to Nainital with a visit and stay in Bhim taal, Saat taal, Naukuchia taal and finally Nainital. Move easy now and you have Darjeeling with it's amazing scenic beauty esp of the sunsets and sunrise and the mighty snow laden peaks including a vote of the mount Everest. From there you could go to Sikkim and the trail is endless to experience the mountains.
The thing about traveling in May is that all the mountains and the passes are open to you while the season is still fresh and the monsoons still a month away which makes it the ideal month to experience and reminders yourself in the pure and beautiful environs of the Himalayas. in case you plan the nilgiris, let me know and perhaps I can provide some tips for that is another experience altogether. Only, this area can be experienced and enjoyed almost any time of the year.