Upon graduation, all students holding an F-1 visa can be authorized for employment for up to 12 months under the Optical Training (OPT) program. Students with a degree in science, technology, engineering, or mathematics (STEM) are eligible for an OPT extension of up to 24 additional months, making this a maximum of 3 years of "work permit" for STEM international students. Non-STEM students only get 12 months before the risk of falling out of legal status. J-1 students in many cases, get up to 30 days only.
If your long-term interest is to continue your career and professional development in the USA after your studies, try to avoid institutions or programs that would only give you a J-1 visa. While you may not know the visa you may get at the beginning of your application process, it may be a question worth asking a current student or beneficiary of the scholarship you are applying for in that university. You may also find out from the international student office of the institution as they are usually in charge of processing official admission documents. The F-1 visa is the best form of student visa for a long-term stay.
Once the OPT period ends, graduates must find an employer willing to sponsor them for a work visa (such as an H-1B visa) in order to continue working in the United States. This is difficult for many employers as the US government puts a lot of regulatory restrictions to ensure that employers consider US citizens and residents first. Which, I believe, is fair. This comes at serious costs for employers leading to the unwillingness and hesitancy of many employers to sponsor foreign talents. In addition, it is not absolute that you'd get this H-1B visa even when you find a willing employer to sponsor your H-1b. This is because the number of applications every year far exceeds the limit of approved visas made available by the government. Hence, the final H-1B availability is often done through a lottery system. I have seen cases where someone wasn't picked in the lottery 3 times in a row leading to him having to leave the US and his job after the expiration of his OPT. Do you really want to leave your future in the hands of lady luck?
Apart from the cumbersome employer-sponsored H-1b visa, international students and exchange visitors may remain in the United States if they are eligible for family- or spouse-sponsored green cards and the K-1 visa for a fiancé(e)s of US citizens and some nonimmigrant visas. This is why many scholars--who would have concentrated on scholarship and advancing knowledge in their fields--get distracted to chase other ungodly things, like "falling in love"-albeit deceitfully in many cases- with a citizen for green cards or paying for the marriage-for-papers package.
The best path to Green Card as an International Student
As international scholars, your talents can be enough for you to self-sponsor your green card through the process called the National Interest Waiver (NIW). The NIW is under the EB2 immigrant visa category that basically uses your past achievements and future prospects to argue a case for the green card. Basically, you will petition the United States government, that you are special, and you have future plans (supported by your training and achievements) that would benefit at least one area of the US National Interest, and it would be to the benefit of the US if they allow you to stay and pursue your plans.
The best thing is; it is self-sponsored! So you do not go through the financial stress or emotional rollercoaster associated with the employer-sponsored path or the legal/illegal marriage path. The NIW-green card is not just for you alone, as your spouse, and children can also get their green cards! The NIW to green card process is also very much faster with many people going through all the processes in under a year! In contrast, you have to be on the H1-b for 3-5 years before you can file for a green card. If your employer lets you go during this period, you are on your own. It is what I did, and I have created a resource to teach any scholar that wants to learn.
You may find the resource here. It is my hope that this would help you plan and collate your documents in preparation for the NIW approval.