Welcome to the forum.
Get yourself some of the magazines like Physics Today, Business Schools Today, and similar. These are he "happy gossip" mags, telling things like what profs have been hired, what labs have expanded, and things of this nature. Your university librarian may be able to help. Possibly Google.
Find the issues that show recent grads and where they got jobs. Look for the ones that got jobs that make you want to emulate them. Possibly a given school will have this info on their web site, to some extent any way. That will give you some schools to think about.
What degree you can take is strongly a function of what school you can get to accept you. Find some candidate schools and contact them. Google them and then get your email working. Find out what their admission requirements are. Also be sure to ask about scholarships. Many scholarships require you to apply. You do not get considered unless you apply.
Whether you take an MBA or not depends on your desired path after that. An MBA probably means you want to go into the business end of the universe, probably some sort of management. Possibly start your own company. This is as opposed to technical or theoretical work. If that is your goal then an MBA may be for you.