Poverty
Sadly, much of India remains desperately poor. You will encounter poor people nearly everywhere, and see some truly miserable living situations. It's not uncommon to encounter poor people begging right outside luxury 5-star resorts - or to be able to see slums looking out the windows of such resorts. Be prepared for this. People will see you and will ask you for money. Even if you're sitting in a taxi in traffic, or walking down the street. It's up to you how to respond, but bear in mind a few things. If you are seen giving someone money, you may be approached by many others leading to an awkward situation or worse. Second, some beggars will do dangerous things like send children into traffic to beg - and you may not want to support this. Be careful.
Crime (Especially against Women)
While India is generally pretty safe - it does have definite problems when it comes to the safety of women. Think hard about going to India if you are planning to travel alone as a woman. Everyone should be attentive of their surroundings at all time, avoid becoming overly inebriated and generally use the same safe behavior as elsewhere in the world. Watch out for pickpockets, and try not to wear showy jewelry or show lots of money. Keep that phone tucked away, or firmly in hand if being used. Watch out for moped drivers trying to snatch purses or other items from pedestrians.
Cleanliness
Like many places outside the western world, food hygiene especially can be an issue. It's not uncommon to get low-level intestinal issues like mild diarrhea when traveling in India - and it's possible to get much, much worse afflications. Be very careful what you eat. Use only bottled water with sealed caps. Don't eat food that has not been cooked, unless you are eating the insides - for example bananas. Freshly cooked food is generally safe, even from street food vendors. Make sure it's been boiled, fried or otherwise exposed to heat. High end restaurants in resorts will generally be safer.
Traffic
Traffic in India can be crazy. They also drive on the other side of the street. And throw into it the fact that drivers may encounter cows, camels, horses or other animals on the road! Be very very careful when walking on the street and especially when crossing. Traffic is very heavy and chaotic, with many tuktuks and mopeds weaving in and out of traffic. Watch out!
Cons and Scams
In general, Indians are extremely friendly and helpful. But as in any place with lots of tourism, there are those who prey on tourists. The best way to prepare for this is to read up on your destination - most travel sites will tell you specific scams to watch out for. Be especially on-guard anywhere that has lots of tourists - airports and site-seeing destinations are common hunting grounds for the scammers. Tip - Uber works in India in many places, which can save you from having to deal with taxi drivers and using cash...