India is one of the most linguistically varied countries in the world and many Indians are multi-lingual. Even on the village level, it is common to encounter people speaking two, three, four, or more languages. And English, of varying quality, is often one of them.
Very basic phrases
It is quite easy to travel in India speaking only English. (Your English can be very basic, but you should have some ability. Even in former European territories, like Pondicherry or Goa, very few people speak the colonial language.)
However, making an attempt, however limited, to get around in a local language can have a magical effect and can enhance your India experience. When you can say " No " or ask directions in a local language, you're demonstrating your interest in the local culture and implying local experience. This earns you respect and makes interactions easier. Of course, it also creates the assumption that you speak more of the language than you do. One of the most useful phrases after "I don't want any " may be "I only speak a little. "
The simple words suggested here, and the phrases you can build from them, are so basic that you'll be speaking only a "pigeon" form of the language. You will only be able to make a request or ask a simple question. You will not be able to interact with free speech. However, these words would get you started and you can easily add to your vocabulary as you go along. This is just to get you started.
Because in India so many English words have entered local languages and are commonly understood it is possible to learn some basic structure and local verbs, then use these with English nouns. The advocates for purity of various Indian languages may be justifiably outraged but it will help you get around. Later, if you have on interest in learning more there are books available everywhere. Local people will be quite helpful, both in correcting your pronunciation, if asked, and helping you add to your vocabulary.
For the South, I've given Tamil and Telegu for now, (Kannada and Malayalam are coming soon) and for the North, Hindi. Hindi you can use over much of India, but it is not so useful in the South. If you have time to learn these few words in only one language, concentrate on Hindi, if you will be mostly in the North and Tamil if you are mostly in the South. Since the four South Indian languages are related, I find I can use this basic Tamil in Kerala, and places with large Tamil populations, like Bangalore. Once you learn these words in one southern language the others become easier because of the many similarities, especially at this basic a level.