In today's society, what could we learn from the Romans during the collapse of the Republic?
There were several linked causes to the fall of the Republic. Perhaps the key cause was rising income inequality -- a cause immediately applicable to the United States in which income inequality has increased dramatically over the past 20 years.
During the years from 300 BC on, aristocrats (Senatorial class) gained control of long term leases over public lands in Italy, often driving out small farmers, and evolving a system called latifundia of enormous plantations depending on slave labor.
The expansion of the Roman empire fuelled inequality. The property qualification meant that free farmers could be conscripted into the army for long terms of service as foot soldiers. To avoid this many abandoned their farms and moved to Rome, increasing the concentration of property into large estates.
Politically, the reforms of the Gracchi gave increased power to the plebians. Warring political factions would bribe the plebians (especially the restless landless mobs of Rome) with 'bread and circuses' to buy votes and prevent riots. Only by increasing conquests and exploitation of the provinces could this economic model be sustained.
Modern politicians might learn lessons from the problems of income inequality, from economic growth dependent on using cheap foreign labor and materials (Roman provinces structurally similar to outsourcing to developing world), bribing populace with bread and circuses vs. education, and factional strife as a political modus vivendi.