how did Roman government win support of conquered people?
This is a great question. The first point to keep in mind is that the Roman civilization lasted for a very long time. Traditionally it started in 753 BC and lasted to 1453 AD (at least the Eastern Empire). So, we need to ask what time period we are speaking of. So, I will limit my answer to the Republic and Early Empire.
During this time, the Romans got support of conquered people in two main ways. First, Rome could give conquered peoples and lands certain amount of political privileges, such as military support or even citizenship to a certain degree. To really loyal cities, the Romans may even give full citizenship, which meant the ability to vote. One of the great qualities of the Romans was their ability to assimilate people into Roman society.
Second, the Romans gave religious freedom. They were a polytheistic state. So, there was much freedom in this area. The Romans rarely imposed in this area.