The Easy Part.
You have the Internet! You can research places, slang, and to some extent the culture of Americans, or British, in order to write your story. You have travelogues, you have some American comedy and talk shows (to show you what Americans find funny or enjoy talking about), you have American blogs, American twitter, American YouTube, comment areas on Huffington Post and other sites; even sites like this to pick up speech patterns and American slang.
The Hard Part.
A plausible American / British attitude (For people raised British or American) toward life, risk, sex, romance, work, possessions, their standard of living, what they tolerate in culture you would shun, what they shun that you would find inoffensive.
I can think of a few imperfect solutions for this dilemma:
1) Read. Read analytically other fiction with American/British main characters, and pay close attention to cultural portrayals, especially those that seem foreign to you. It may be things they don't care at all about (e.g. homosexuality, atheism) or things they DO care about that seem wrong to you (eg. gun rights for Americans, respect for royalty (or hatred of it) for British.)
2) An American raised in India. The child of an ambassador, for example. They can have a mix of cultures.
3) Sidekick POV. Add a minor Indian character to the crew, and tell the story through their POV. Much like Dr. Watson was necessary to the Sherlock Holmes stories, told through his POV, even though the hero was always Sherlock. As an Indian you can tell the story through an Indian POV about a foreign crew in a foreign land. Then you are "insulated" against some misinterpretation of the foreign culture.
None of these are perfect, it can be difficult for anybody to properly portray a culture they did not grow up in. Particularly in regard to sensitive cultural issues: Racism, sexism, homosexuality, religion, women's rights, employment rights (like unionization), what the law allows (business is more highly regulated in the USA than India; e.g. with sexual harassment laws, equal rights laws, etc), including in the USA what people can sue each other over.