If we get married in the states, should we immediately apply for a green card for him?

You may be able to sponsor him. Consult with an attorney.

Generally, when a Lawful Permanent Resident becomes married to a foreign national in the U.S. while the foreign national is lawfully present with a nonimmigrant visa, the Permanent Resident may sponsor the foreign national in the Family-based Second Preference visa category. According to the U.S. Department of State's October, 2017 Visa Bulletin, there is a significant backlog for visas in this category (for filing purposes, visas are available for those with a Priority Date of November 1, 2016, or earlier, and for final action date purposes, visas are available for those with a Priority Date of October 22, 2015, or earlier). This means that it will not be possible to file an Application for Adjustment of Status (with applications for an Employment Authorization Document and for Advance Parole) concurrently with a Petition for Alien Relative. Generally, it would be necessary to file the Petition for Relative Alien upon becoming married, and then either (1) consular process from abroad, or (2) if the foreign national spouse otherwise has a visa authorizing him to remain in the U.S. until his Priority Date is reached, apply for adjustment of status in the U.S. when his Priority Date has been reached. If you may become eligible to apply to become a naturalized citizen soon, other considerations should be taken into account. Beyond this, there are many other details that determine immigration eligibility. There really is no substitute 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.