I have been curious about the evolution of the collected history of rome. I'm familiar with Livy but not many of his contemporaries, or those who came after him. For the history that's inaccurate, how do we know it's inaccurate? For the history that's accurate, same question.
In essence, what I'm wondering is the history of the history of Rome.
We are lucky to have quite a few texts that have been preserved or rediscovered that give us a good look at Rome.
These include, and are def not limited to:
Sallust (famous for his retelling of the Caitiline conspiracy)
Suetonius, more of a biographer than an historian. Wrote the fantastic the Twelve Caesars. He had access to loads of imperial files and letters, as he was in the court of Trajan.
Pliny the Younger, who wrote extensivly. Unfortunately, not much of his work survives.
Cassius Dio, whose work we have thanks to third century Byzantine monks. It's useful and worth a read
Tacitus, the almighty cynic, who wrote The Annals of Imperial Rome, a fantastic text on Rome from under Tiberius to Nero ( a span of about 50 years ). His writing is, like all the above, written with an agenda. It serves as a clear warning of the dangers of complete power, and is responsible for much of our somewhat bias view of Nero. This leads on to the next part of your question.
Accuracy. It is often difficult to track the accuracy of ancient sources. Often, there writing is thematic: themes include good vs evil (Tacitus' Thrasae Paetus vs Nero), and moralistic (again, Tacitus often includes or omits details to push his agenda. Often historians record speeches in a inaccurate way- they spice them up to make them more interesting.
We can attempt to fact check a variety of ways. Coins can sometimes support the text. An example of this is the many coins we find from the beginning of Nero's rule that show him and his mother Agrippina. Suetonius, Dio and Tacitus all talk at length of their close relationship, and these coins support this. Likewise, we can verify how far the romans may have expanded into various territories mentioned in the texts by examining archeological dig sites for proof.
Finally, we can attempt to cross reference the writers with each other. If you're really interested in this, I cannot recommend Mellor's The Historians of Ancient Rome enough, it covers it much better than I ever could!
Edit: Sorry, I really struggle with laying out this replies and understanding the formatting of bullet points, despite reading the guide!