These are some really big questions you have here. I’ll try to be concise, but books have been written about each one of these topics.

Governance differences between the Roman Republic and Roman Empire

Basically, governance in the Republic was (partially) based upon the election of officials and checks and balances, while in the Empire it was based upon inherited power in one individual.

The Republic was governed by the Assemblies, the Senate and elected magistrates. Assemblies could vote on legislation and officials, the Senate gave advice, and the magistrates had various functions.

Assemblies voted for different magistrates who had a variety of duties which evolved over time. Normally these officials served for one year and most of them had at least one colleague in office. These magistrates included the censor who was in charge of morality, citizen lists and who was qualified to be in the Senate. The quaestor handled some finances while the aedile was in charge of public infrastructure and festivals. The tribune could be a military office or else could be a Plebeian Tribune. The latter could propose legislation to the Plebeian assembly as well as veto legislation. The highest office was the consul. The consul had extensive executive, legislative and judicial powers, as well as the power to command armies.

There were many other offices. The pontifex maximus was the chief priest and served for life. Religion was extremely important to governance in Rome, so this was an important office. The dictator had special powers, normally for a very short time. Sometimes a dictator was appointed for very specific and limited reasons, such as holding elections.

The Senate actually had very little legal power. However, this body had enormous authority and soft power. The Senate issued ‘advice’ which was usually followed. This body was made up of several hundred individuals. Typically after a person had served as a magistrate he joined the Senate.

In the Empire almost all power was held by one individual, who we call the Emperor. Interestingly, what happened formally is that the Emperor was given the power of the different magistrates. For instance, he might be given consular and tribune power (among many others). He had the power to command armies, ‘propose’ legislation, act as judge and determine who could serve in the Senate.

So on the surface it looked like the old system of government was continuing as before. Magistrates were still elected, the Senate still debated and gave advice, and Assemblies continued voting on legislation. But make no mistake: the Emperor called the shots.

The Emperor also tried to choose his successor. Sometimes this was a natural son, or sometimes this was someone adopted.

Social differences between the Roman Republic and Roman Empire

I honestly don’t know how to address this question. The Republic and Empire lasted hundreds of years and there was great evolution over time. If you have specific questions let me know and I’ll try to answer.

Caesar and the Ides of March

According to the (perhaps apocryphal) story, a soothsayer once told Julius Caesar to beware the Ides of March (15 March). Caesar met the soothsayer that day and teased him about it being the Ides of March but nothing had happened. The soothsayer replied that the day was not over yet. Later that day Caesar was assassinated in the Senate.

The reason that Caesar was assassinated was that various Romans were concerned that he was seizing all power and might even want to be crowned king.

This was just one episode of a very long process. For several generations the struggle between powerful individuals had become, at times, violent. It was going on before Caesar and went on after him.

Legislation in the Roman Republic

Legislation could be proposed by various magistrates. Assemblies were formally convened by magistrates and voted on the legislation put before them. Generally there was no debate – that took place in the Senate.

There were several different Assemblies, like the Comitia Centuriata which was organised by ‘centuries’ based upon wealth and the Comitia Tributa which was organised by ‘tribes’ based (originally) upon geography. The Concilium Plebis was a body made up of the plebeian class (as opposed to the patricians).

The fall of the Republic and the transition to Empire

It would probably be easier to explain the meaning of life. Basically, Roman society and the Roman system of governance was no longer stable. This instability led to a prolonged period of internal strife, which led to the introduction of a new system. Note this new system was introduced gradually over time and continued to evolve. There is not a clear division between the two.

For more specific information I encourage you to take a look at this old AMA post handled by some of our best contributors: The Decline and Fall of the Roman Republic

Addendum By necessity most of my answers were extremely general and glossed over many aspects. Some of these aspects are quite specialised and arcane. Feel free to ask specific follow-up questions. I’ll answer if I can, and hopefully some of our specialist experts can help if necessary.

Also, this statement of yours deserves a mention:

Rome really interests me, but I really don't want to read a giant book about it.

I hope you change your mind. Check out the subreddit’s book list