Regarding the level of education, it depends on what field and how much work experience you have.
Regarding the process, first you should change status to H-1b. Your employer will need to file a Labor Condition Application (Form ETA 9035) with the Department of Labor (DOL) to prove that your employer needs to hire a foreign worker and that your employer will be paying you the "prevailing wage" for that type of work in that geographical area. Fulfilling all the requirements and obtaining a certified approval for the Labor Condition Application from DOL takes about a month. After that, your employer can file an H-1b petition (USCIS Form I-129) for you, and you'll probably want to pay the $1,000 fee for premium processing, which will give you the approval within two weeks if everything is correct in the petition. If you don't do premium processing, it could take from 4 to 6 months to obtain approval of the H-1b petition. In any case, make sure your petition is filed with USCIS prior to the expiry of your F-1/OPT to ensure that you can continue working legally in the United States. Once the petition is filed (meaning USCIS has issued a receipt), you can then move to a different employer but that new employer will have to file (meaning USCIS has issued a receipt) a new petition before you can leave the first job and start with the second job. Also, departure from the U.S. during pendancy of an H-1b petition constitutes abandonment of that petition.
This process is required because you're currently not in H-1b status. You're permitted a total of six years of H-1b status in the U.S. prior to having to return to your home country for a year before being able to return as an H-1b again. However, if a Green Card Application is pending, then your H-1b status can be extended for one year at a time every year the Green Card Application is pending.
If your employer is willing to sponsor you for your Green Card, your employer will have to file a Labor Condition Application (Form ETA 9035) again with DOL. This usually takes 4-6 months for approval but can take even longer. Once the Labor Condition Application is approved, your employer can file an Immigrant Petiton (USCIS Form I-140) with USCIS, and if the priority date is current for your preference category, which is based on your educational and work experience and the education and work-experience requirements for the position, then you can concurrently file a Green Card Application (USCIS Form I-485). If the priority date is not current, you'll have to wait for it to be come current prior to filing the Green Card Application. If the Immigrant Petition is approved and the Green Card Application has been pending for more than 180 days, then you can change employers without having to have filed a new Immigrant Petition or having to to file a new Green Card Application provided that the new position is in the same or similar occupation and field. If the Immigrant Petition has not yet been approved but the Green Card Application has been pending for more than 180 days, then it becomes trickier, but if you're confident your former employer will not withdraw the Immigrant Petition, then provided the new position is in the same or similar occupation and field, you could change employers without having to have filed a new Immigrant Petition or having to file a new Green Card Application.