So we've got gibbons history of the decline and fall of the roman empire for a beautiful history of the empire, but what is it there for learning about the history of the roman republic?
Plutarch
>>8408569
SPQR: A History of Ancient Rome by Mary Beard is a good quick read covering the history and historicity of the founding of Rome to the late republic and early empire.
The Catiline Conspiracy is briefly examined and holds as a kind of binding moral through the book, and if you know latin literature you won't miss any references.
>>8408569
I don't know of any secondary sources that match the fame of Gibbon, and so much of the early history (founding to the Gallic sack) is basically unrecorded and left to legend that it's hard to write a decent history of it.
With that said, there are some great primary sources on the monarchy (753-509BC) and the republic. Livy is the most popular source for the earliest years, and is only one of two that has a continuous narrative of the monarchy and the early republic. The other is Dionysius of Halicarnassus, who is okay and has some interesting moments, but largely lacks sense of proportion, spending too much time on certain things, and ultimately ends up bogging the story down with tons of rhetoric which, from what I've read, is largely ripped off from earlier rhetoricians. Also he's only published in 7 Loeb volumes which will run you about $200 total.
Polybius has a great history of the Punic wars. He's not as "exciting" as Livy, but is a better historian and a great teacher of history. Also provides a wider perspective on the Punic wars, since only Livy's account of the Second war exists; the whole range of Polybius exists, although much of it is fragmented.
Appian is like a worse Polybius with a broader focus, and his earlier books (not the commonly found ones on the civil wars) are a great source on the Republic. He's the only complete source for the Third Punic war, and certainly has his moments as a writer.
Sallust's histories only explicitly talk about very narrow events of the late Republic, but he's indispensable in capturing the political and social zeitgeist of a Rome corrupted after the fall of Carthage. Very short, extremely fun, well worth the read.
Plutarch may fill some gaps in the history of Rome before the empire, but it's often debated whether he's a "real" historian or just a storyteller. Definitely worth checking out, but back him up with some other sources.
Dio Cassius also has some existing, but fragmented, books on early Rome, but they're not great and can't really be read in place of a cohesive, if fictionalized, account like Livy.
Also Diodorus wrote at the very end of the Republic (~60-30BC), but is basically a Greek historian. Very little about Rome, and for good reason: He properly contextualizes a Rome which only recently had become important on a world scale in a hitherto Greek world. Not worth reading as a Roman history beyond assigning Rome in her place as I just described.
>>8409182
Classics freak here, I more or less agree with everything this anon said. I would throw into the mix Cicero's orations and letters. If I could just pick a few speeches, the Pro Sestio, the In Verrem (skim this it's like 400 pages but important for understanding how fucked provincial administration was), the Pro Clodio, the Catilinarians and the Philippics. That's passing over a lot of great stuff, though, because Cicero was a genius and everything he said/wrote is worth reading.
His letters, in particular, are more of a primary source than anything else that survives. Get them in a chronological edition with notes (the manuscripts have the letters all out of order and there are separate manuscript traditions for different recipients/classes of recipients etc.)