What is the process for marrying a non US citizen who is currently in the US legally with a temporary visa?

The processes for obtaining a marriage license and completing a wedding are governed by the laws of the state where the marriage takes place. Usually those processes and rules (including ones about needed documents) are fairly straightforward and can be found in state or county government websites. Becoming married to a U.S. citizen will not, alone, entitle your fiance to remain in or be authorized to work in the U.S., but marriage can create eligibility for a marriage-based adjustment of status application - a process by which your then-husband would become a Lawful Permanent Resident (would get a "Green Card"). Of course, there are many other details that determine eligibility for the adjustment of status application process beyond merely becoming married. It would be wise for you and your fiance to consult with an immigration attorney who, after learning all of the relevant information, could advise about eligibilities, options and strategies to attain your goals, and who then could offer legal representation in the often complex application process.