I'm reading Collar and Daniell's First Year in Latin right now and they mention that Latin has no articles such as "a", "an", and "the". Is this true? I have heard the book be inaccurate before.
The book is correct. There is no equivalent to "the" in Classical Latin.
In Vulgar Latin, the demonstrative ille (which means "that" in Classical Latin) got bleached into a definite article, with a meaning similar to English "the". That's where forms like Spanish el, Italian il, French le, and so on come from. But that wasn't good Classical style.
As discussed here:
the word "ly" is occasionally used as a definite article in mediaeval Church Latin.
There is no article in Latin. You just don't translate it. "The girl is the servant of the lady" = "puella est ancilla matronae".
However, there are some cases where an article may be translated with various words: this happens when you're not really using "the" or "a" as an article, but to mean "the famous", "that famous", "a certain". For instance: "the teacher is wise" = "magister savius est", but "the famous teacher is wise" = "ille magister savius est". "A certain Flavius" may be rendered as "Flavius aliquis", literally "someone Flavius". This is especially true with proper names: "ille Cato, censor perclarus". This in particular explains the "Winnie ille Pu" example.
The other answers already covered the most important part of your question but since you did mention
you do know those are just archaic forms of "one" and the way to express that is unus, una, unum, right? Similar to Chinese, though, Classical Latin can express this idea but usually just doesn't bother to. Use unus &c. in places where you might say "one" or really want to emphasize the "a" (eg, when you'd pronounce it /aɪ/ instead of /ə/). More often, rather than repeat unum every other sentence, Classical Latin depended on context or synonyms like ullus/a/um ("any") and certus/a/um ("a certain...").
In lower class/vulgar/medieval Latin, of course, unus/a/um became more and more common, which is where you get French un(e), Spanish uno/a, &c.
The theoretical proto-Indo-European language that is at the root of all Western (except Basque), all Near Eastern and some Middle (eg Farsi, Sumerian) and South Asian (eg Hindi) languages had no articles so they tended to adopt them from the native cultures the speakers conquered. Latin, as stated above, had to form its own, possibly from the influence of their neighbours whom, as non-native speakers, would try to look for an equivalent.