For example, historians will often quote or paragraph what a Greek philosopher said. But the citation is usually to another text book or a translation (often in a compilation of translations).
How does one determine if these sources are accurate and reliable?
For specifically quotes, what one does is try to figure out what the original source is or might be. So you can "leap frog" backwards through citations. Found something that doesn't have a citation? That's either the end of the road, or you look for even earlier versions elsewhere, but that's much harder. Fortunately we do live in a world where a great many things have been made digitized and searchable through numerous databases, so that greatly aids in it.
Sometimes one ends up finding one's way back to something that looks like an original text. Sometimes one finds a dead-end — but even the dead-end can be useful, if the dead-end is something known to you, or suspicious in its own right. So, for example, I occasionally look into putative quotes assigned to people like Albert Einstein. If you trace this back to something contemporary with Einstein, or a letter by him, then great, you've got something plausible. If it only traces back to popular books from the 1980s and no earlier — it might be wholly made up, though it depends on who the author of the original text with the quote is (e.g., several quotes trace back to recollections by Abraham Pais, who was a friend of Einstein's, so it's more plausible that they are based on something Einstein said).
Sometimes you only can get so far, and you just indicate in the footnote that's how far back it went, if you want to use the quote. I use a quote from Leo Szilard in my book ("The SECRET stamp is the most powerful weapon ever invented") that I really loved. I could not find it written in Szilard's hand, though; but I was able to trace back to the first person who said Szilard said it (a physicist named Ralph Lapp) and judged that "good enough" to include (with the proper citation), since Lapp was a friend of Szilard's.
There are always going to be areas of judgment and uncertainty, especially if you are looking at something that is cast back across millennia. The role that the quote places in your work can also determine how much you care about the provenance; if it's just a bit of color, or agrees with many other independent sources, then one's bar of evidence and burden of proof is probably less than if it is the kind of statement that the entire paper hinges around.
I’d direct you to this section of the FAQ: Historiography and studying primary sources
Specifically: