Rome is such a big topic, it not only deserves a post of its own, but it can be studied in isolation as it encompasses so much of Western history. The most obvious reason for this is the Romans were around for a long time. The traditional dating of the start of the Republic is 509 BC and the end of the Western Empire is 476 AD. That's roughly one thousand years and it was an action-packed and dramatic thousand year run too. Rome features some of the most colorful characters in human history.
There is something else. The Greeks were a culture, but the Romans were a civilization that incorporated the culture of the Greeks. Culture is the spirit of the people, the worldview that results from their shared history. The Greek culture was a product of the people and their place in the eastern Mediterranean. It is their literature and philosophy, their understanding of man's relationship to other men and the world in general. It is the structure of their language, which is a reflection of how they thought about the world.
Civilization, on the other hand, is the end point, the destination of culture. It is the tangible product of cultural yearnings. Culture is the belief in self-governance. Civilization is a constitutional assembly. The Romans adopted Greek culture and built a mighty civilization. Because Rome carried the spirit of the Greeks throughout the known world, planting its seeds throughout Europe, knowing the Romans is to know the essence of Western culture. You cannot know the West, without knowing Rome.
Traditionally, you learned about Rome by reading Gibbon. You can buy The Decline and Fall of the Roman Empire from Amazon for $133.00 or if you want to read it from a tablet, you can get it for two dollars, which is one million Canadian dollars.There are many better options than this, which we will get into next, but having some familiarity with Gibbon is a good idea, as you will find that later writers refer to him regularly. Spending a couple of dollars to have him in your digital library is a good idea, even if it is just as a reference work.
Then there is Latin. Learning some Latin is a good idea for anyone who likes knowing stuff. One of the worst things to happen to education in the modern age is the end of Latin instruction in the schools. Knowing even a little bit helps expand your language skills as an English speaker. In the case of Roman history, it is a good idea to have some basic knowledge of Latin, which means buying a primary school grammar used by homeschoolers. You are not going to be reading Caesar in the original, but you get enough of the basics.It's not necessary but recommended.
The most basic way of thinking about Rome is to break up its history into three periods. The founding through the Republic, the Empire starting with Augustus and then the late Empire, when things got squirrely and the Empire was in decline. A recent book that will take you from the founding through the Empire, but stopping at the Late Empire, is Mary Beard's SPQR: A History of Ancient Rome. It was a best seller and it a great read. An excellent study on the transition of Rome from a Republic to an Empire is The Breakdown of the Roman Republic: From Oligarchy to Empire.