Going to break this down into a couple of sections

1. What the senators and equestrians thought (they were the elites of the roman people, and a huge amount of our knowledge comes from texts written by men from within this sector)

2. What your everyday roman felt. Records are less clear on this, almost non existent actually.

So, various members of the senate and the nobility felt deeply upset about the transition from Republic to rule by a single man. After Carthage was defeated and sacked in 146 BC, Roman elites like Cato the Elder were already angered by what they saw as the erosion of traditional Roman values due to the huge amount of money coming into the capital. They believed this wealth was ruining the ability of the senators and equestrians to behave as they should be.

Fast forward just under a century, and you have Julius Caesar being assassinated in 44BC. His assassins claim to have feared he was going to become a dictator or ruler for life. (Romans hated the idea of a monarchy). Combined with the rise in warlord generals, who could command multiple legions of men loyal to them, and it was clear that the republic was falling apart.

2. Ordinary people. Suetonius mentions that they were just really really war tired by the time Octavian deftly steered the state from Republic to empire. Civil war had raged on and off for a century, and now they had a chance for peace. On a local scale, government continued somewhat normally, no huge drastic changes.

Hope this somewhat helps, very vague and feel free to DM if you need more info!