Yes. Currently, a U.S. citizen can apply for citizenship in another country (i.e. Naturalization) without losing U.S. citizenship. A person with a US citizenship can become a citizen of another...
If both your countries of nationality require a visa to enter the destination country, then you should obtain a visa and have it stamped in one of your passports prior to your departure. If the...
India does not allow dual citizenship. But you can apply and get a passport-like booklet, at the nearest embassy and consulate general, that will allow you to travel to India visa-free.