How did Rome handle the difficulties of governing its large empire?

Rome didn't really handle the problems, it was too big for its own good, it did well under Julius Caesar, but after his death the families vied for power over Rome. It did its best, increased policing by custodes (Roman Guards). The Roman Empire made good use of its legions, but due to its size it survived in different forms after Rome itself had fell. I.e. Byzantine Empire the birth-place of Christianity. Mass conversion of conquered territories known as "Romanization" allowed them to gain Roman citizenship. Roman leaders would be set up as a localized government of conquered regions. They were Rex Roma, or Roman Kings acting as multiple heads of a hydra for the Roman Empire. Caesar Augustus also created amazing policies to unite the empire, thus bringing peace and harmony to Rome for a few years.

These rulers were given a measure of local autonomy, power over the native inhabitants, and even permission to maintain armed forces. But Rome kept a close eye on any possible usurpers to prevent an autocracy.

Rome's army was unmatched and could crush any rebellion with brute and violent force. Rome had an order of executioners. They were the Carnifex, they executed, tortured and worked with custodes to capture and publicly torture possible rebel leaders.

Rome was an empire of unmatched military power, wealth, and art. Yet, due to the corruption within the political housing, even the Praetorian Guard, right hand of the Emperor took bribes by usurpers to turn the other cheek on assassinations, letting assassins gain entry and kill those they were supposed to protect.