Depends on what you mean by "average." During Augustus' reign there were over 50 million people living within a sprawling empire that spanned centuries in time! I think I know what you are asking, though forgive me for answering broadly. As you may know, most people for most of European history were involved in agriculture. The Roman Empire had a remarkably high urban population for a pre-industrial civilization, but the vast majority still toiled away in the fields.
Many farmers would live near some form of urban settlement so they could sell their produce, whether it be a port or town. As Rome had a very healthy trade economy for much of its history, news would often travel by traders. Thus you may learn a good deal from your visit to the small port or town near your farm. Of course information could be quite dated by the time it reaches you depending on where you lived in the empire. Furthermore, news could have been passed from person to person many times and the original message could be distorted. Those in the far reaches of the Roman Empire would sometimes simply hear nothing. Supposedly many peasants did not even know the Western Empire had fallen centuries after it had. They just continued paying taxes to the collectors when they came by.

As for the urban Romans, their were many sources. Orators would yell news outside civic buildings, gossip in the bathhouse, news from traders etc. Celebrations would occur in major cities to celebrate new emperors or military victories.

Keep in mind that Rome had an extensive bureaucracy to match its extensive empire. They recorded everything from significant news like a new emperor to boring administrative details like cost of grain. They also had a system to share news with other administrative centers.

For military leaders that required urgent news it would depend on where they were. They would dispatch a messenger by the quickest available medium be it ship or horse. Thanks to dominance of the Mediterranean and the famous road network of Rome, transportation of messages would be expedited.