Would it be easier to get my fiancé a green card if we both go to Mexico and get married?

If you marry in Mexico, he can be stuck there easily a year or two best case scenario. If you marry here, you can be with him during the waiting process. You can always marry again in Mexico once he has his consulate appointment.

By your explanation of the facts, I am assuming that your boyfriend has a visa that allows him to come in and out of the US as he pleases. If this is the case, you can go to Mexico and get married. However, the moment immigration notices he got married, he is not going to be allowed to return to the US until his fianc? visa is available. Immigration is afraid that if he is allowed to enter the US again, he will stay and do the process in the US and if not approved, he may stay illegally in the US. Thus, he will have to wait for the process in Mexico. You will file the fianc? visa and the timing for his return will depend on how fast the consulate in his country and city are working. It should take shortest 3 months, longest over a year. Time with immigration is unpredictable. If you marry while he is in the US, and he entered with a visa, you file immediately for him through a family based visa petition and apply for residency and work permit all at the time. This way, in (about) three months he will have a work permit while the process takes place. Residency should take from 6-8 months (but again, time for immigration purposes is relative and may vary by case).

I strongly recommend consulting with an immigration attorney before your boyfriend leaves the United States as he is likely to trigger a 10 year ban for returning to the country. You should have his immigration history reviewed and options assessed.

If he departs the US, he will be barred from reentry to the US for a period of 10 years because of his unlawful status in the US and that can only be waived by a showing of extreme hardship to a US citizen spouse, which is fairly difficult to do in most cases. If you don't depart the US, however, because of his unlawful entry to the US, he will not be eligible to legalize his status within the US, even on the basis of marriage to a US citizen.

If he entered the US illegally, then he will trigger the 10-year bar when he leaves the US. He will need to file a waiver to forgive the immigration violation. You will need to be married, and then file the I-130 relative petition, and then after it is approved, he applies for the visa at his home consulate with the waiver. There is proposed legislation that would allow the USCIS to adjudicate such waivers prior to him leaving so there is some level of certainty for visa issuance.