Here are a few books about the Roman Empire on my kindle that I either read, skimmed or getting to.
Robin Lane Fox’s The Classical World: An Epic History from Homer to Hadrian (2008). It’s fairly up to date and since most books either focus on Greece OR Rome it’s rare to find one for the period where they coexisted.
Anthony Everitt’s The Rise of Rome: The Making of the World’s Greatest Empire (2013) is an easier read focusing on the leaders of Rome. Everitt also wrote individual biographies of Augustus, Hadrian and Cicero which I haven’t read but I still recommend if you enjoyed his voice and would like to know more about those individuals
While I haven’t gotten far into it, many recommend Mike Duncan’s The History of Rome podcast.
Also SKIP Gibbon’s The History of the Decline and Fall of the Roman Empire. He influenced a lot of later historians but 1. he’s VERY long winded, 2. he doesn’t go into a lot of history that doesn’t support his thesis in the later empire and 3. he wrote it in 1776, so it’s a little outdated
Most of the Kingdom of Rome was mythic at the time Roman historians were writing about it. The main source I can recommend is Titus Livius’ The History of Rome (10). I’d take his history with a lot of salt because he was writing during the early empire and was glorifying the old republic.
But if you want to learn the most about the era, the internet is your friend. This is a nice overview of the era written by a reddit user. Otherwise just surf wikipedia because they also tell you where some sources fall short. In general pages about the ancient unification of Italy will be more reliable than info about the seven Kings. (During the second Samnite war Rome had to unconditionally surrender their entire army because they got trapped in a mountain pass, now why would they lie about that?)
Harriet I. Flower’s Roman Republics was recommended to me because she argues that Rome wasn’t ruled by one monolithic republic but has very distinct republics throughout the era.
Tom Holland’s Rubicon: The Last Years of the Roman Republic (2005) tells how the stage was set for Caesar’s dictatorship.
I also loved Peter Connolly’s The Roman Army (1975). It’s a lighter read (with pictures!) and outdated, but is still a good source on the military through the republican and imperial eras. He also wrote books on Hannibal and Greece.
Many recommend M.I. Finley’s The Ancient Economy (1973) for how the society actually operated (and it’s free on Google books!)
I’ve also been really interested in getting my hands on Quantifying the Roman Economy: Methods and Problems (2009) but it’s a college textbook so I’m in no rush
Late antiquity has been the most fascinating subject for me lately because of how little I know. Peter Heather’s The Fall of the Roman Empire: A New History of Rome and the Barbarians (2007) talks about he various tribes that Rome fought and treated with at the end of the Empire instead of the internal causes of the fall like most other sources do.
Lars Brownworth’s 12 Byzantine Rulers was the podcast that originally got me on late antiquity. A little dry at times, but Brownworth explains how power was shifting towards the East even before the Western fall and a lot of early Christian history for how much Christianity weakened the West, it strengthened the East.
Also, if you have a chance, play Caesar 3 it’s a fun city building game that I played as a kid that taught me about Roman society, the Punic Wars etc.
ADDENDUM This is only semi-related but the biggest thing I hate about the Romans is how they erased the history of anyone the conquered. That’s why we know so little about the other ancient Italian tribes or Carthage. The Punic Wars took around a hundred years and three wars for Rome to finally defeat Carthage, a society that rivaled their control of the Mediterranean yet the only things we know about them are from other Phoenician dig cites in North Africa (They were not a homogeneous culture, closer to Greece’s city states than the Romans)