Yes, you will always need a passport and/or a visa to travel to India. A passport is an internationally-recognised ID issues by the national government of your home country. It is required both for purposes of entry into India and for readmission back into the U.S.A (issued by the embassy in India before you leave the U.S).
The U.S. Department of State’s Country Specific Travel Information assesses safety and health conditions for every country in the world, and issues a travel advisory. Every advisory gives detailed information about potential hazards such as crime, road safety, political instability and terrorism.
This is completely up to you. Many students find cell phones very convenient, others find them unnecessary. Your American cell phone contract will not work overseas without an international plan. Cell phones in India use pre-paid SIM-chips that can be inserted into any phone. If your American cell phone also uses a SIM-chip and has dual- or tri-band settings, you can bring it along and will only have to purchase a new SIM-chip and the pre-paid minutes for it.
India is one of the safest destinations you can consider. With English speaking public easily accessible, travel within the country is never a problem. However make want to take basic safety precautions that one takes while travelling to a new destination. Identify the safest areas to move around in and the most hygienic eateries to explore.