Title says most of it really; but how would an ancient historian such as Plutarch do his research on say, Julius Caesar? Would he search for an older book in Rome? Was it passed down orally? How would they judge what was real and trustworthy and what was myth and legend?
Well they occasionally talk about their methods. Basically there were usually contemporary historians that they would rely on. In your example, Plutarch clearly had Caesar's own writing to draw on. There are only two such sources I can think of now--Thucydides and Caesar/Hirtius (Hirtius finished Caesar's histories). Hannibal brought two Greek historians with him on his campaigns. They do not survive but Polybius mentions them a few times so he must have read them. Polybius also mentions taking a trip to view an inscription Hannibal left after descending the Alps. Livy in turn used Polybius for much of his histories.
Suetonius and Tacitus, the imperial historians, did likewise but they freely admit that their historical sources are atrociously biased. Thus Suetonius often comments on graffiti he has found on Rome and stories he has heard. Along with doing research in correspondence and such.
In Rome much history was maintained by families. At a funeral a prestigious member of the deceased family would give a history of the families accomplishments. This probably influenced Roman historians like Polybius who was a client/hostage of the Scipiones (and wrote about the two biggest men of that family). A similar source would be historical drama. A few historians think that Livy may have been influenced by dramas because so many of his scenes are "dialogue heavy".
Romans (and other Hellenized people) did also keep archives, but they were vulnerable to aging and destruction. The Capitol was destroyed during Cicero's time and took some records with it, and the fire of Rome under Nero was certainly not great for historians. The decrees of the Senate and laws of the People were also usually displayed in inscriptions in public so a person could see them in the law they were researching was still "on the books." In Greece we have a law written in the 8th century BCE (well a 4th c. BCE copy of one at least. It's from Thera giving the decree to found Cyrene, a colony in Africa), so it is in the realm of possibility that the Romans could have kept their inscriptions that long too.
All in all, I would say that the best Roman era historians were as fastidious as modern historians given the sources they had and their access to them. However many of those sources were biased and the goals of history in the ancient world were very different than today. History was a rhetorical art, and rhetoric was the art of convincing and molding opinions. Thus, much like unscrupulous (but still successful) lawyers they used those less-than-ideal sources to augment the good sources for the sake of their argument.