If you’re planning on seeing wildlife, we recommend you travel between October and April. This is the best time to see numerous species, from tiger and unicorn rhino, to spotted deer. However if you have to travel in June, a wildlife safari will be possible. Activities are based around the early morning and late afternoon, when the animals are most active, so your time outside of this can be spent at the lodge pool, or in your fan-cooled / air conditioned room. You will still get hot.
For the cultural sights, the same applies. You can visit the Taj Mahal in Agra, explore the sites of Old and New Delhi, or head to the erotic temples of Khajuraho, but we recommend this in the morning and afternoons to avoid the heat.
The nicest places to travel in June are the southern hill stations, and the far north, the Himalayas. Each of these areas give you a slight respite from the heat, and some absolutely astounding views of the natural world. We suggest also heading to Ladakh a complete contrast to the rest of the country, which is great between June and September.