Dear Tell Me Everything:
This is a very broad question regarding ways to make a career in the US, assuming we are talking about someone coming from India. Reams of pages have been used to write the laws, multitude of legal books have been written for their application, and thousands of attorneys and counselors make a living from the general knowledge and nuances of the rules and regulations. All in all, better people than us can write encyclopedias on this topic. We shall however make an effort to offer some basic information.
The US immigration policy is based on two guiding principles: family reunification and procuring talent that is not readily available in the US. The family reunification principle allows the US citizens to "sponsor" their immediate and extended family members, within certain guidelines. Once approved by the US Immigrations and Naturalization Service (INS), these family members can move permanently to the US. Upon arrival, they can engage in career, education, entrepreneurship, and so on, without any restrictions.
The US also brings in talent on temporary work permits. You might have heard about the H1-B visa category, a temporary work visa category for highly skilled talent. There are other visa categories as well, for which there are annual quotas. Bringing in workers from other countries is based on the determination by the US Labor Department that such individuals would not displace US residents (citizen or permanent resident) from the work force. Accordingly, the US employers are required to demonstrate that they sought to recruit a legal US resident for the job but were unsuccessful. This is done individually for each non-resident. Thus it is not trivial for a non-resident to be recruited by a US company. Furthermore, even for the non-resident talent, the US companies prefer to recruit those non-residents who are already being educated in the US, namely the international students in the US.
Under a temporary work visa, one has a start for a career in the US. For certain types of temporary visas, such as H1-B, it is relatively easy to get them converted into permanent resident status. The process requires considerable time and paperwork, but it is generally doable.
That brings us to the question of education. You inquired about GRE (Graduate Record Examination), an entrance exam typically required for admission into graduate programs (Master's or PhD) in the US. You may find useful links on iMahal:
GRE. Keep in mind that getting admitted to a US school is not sufficient to secure an international student visa. You must also be able to demonstrate your means for financing your education and stay in the US.
We are aware of multitudes of "coaching programs" in India to help one prepare for the entrance exams, such as GRE. You can easily find them advertising themselves in local newspapers. Then you need to put in your own effort to research and select prospective colleges. No particular "institute" in India can provide you the insights and help you need. You need to make your own choices. You need to do some hard work yourself.
Good luck!