For instance, if someone gave me a piece of paper saying: “My name is Joanne and I live I the year 200 AD, and I see dinosaurs running all over the streets all the time”
Or a better example, a book that claims to give details on the events that happened in the past, how do I determine how historically accurate these claims are?
Short answer: information evaluation / source criticism.
The exact way this works is a serious and complicated question which doubtlessly has been extensively answered before. The essence is that there are certain “rules of trade” and methodology historians use.
Let’s take your example of the paper.
Firstly: what is the document made of? You say it’s paper. Now, while true paper was developed around 200 BCE in China it didn’t reach Europe where a Joanne could live until medieval times. So the documents probably much younger.
Secondly: the fact that the document is much younger than claimed is corroborated by the name that is given. Joanne is a Derivate of Hanna (a Hebrew name) which is popular in France and England - but in languages that were not around in 200AD.
Thirdly: the term dinosaur and the identification of scaled (or feathered) reptiles of great size as them is fairly recent. The first dinosaur related description of bone fragments was by Plots (in The Natural History of Oxford-shire (1677)). But he concluded it was a Titan out of Greek myth. Dinosaurs themselves are a product of 19th century research - so...
Therefore the dating giving in the document is dubious. A woman named in a way that would not be used for centuries writes on material not available and using a description not present for 1600 more years.
Not only does this discredit the dating, but also what she says. If the paper could not have been written earlier than the 19th century than it would be 1600+ years from the time it supposedly should tell us something about.
That means it is (a) not a primary source and (b) is quite removed from the original in time (so a secondary source of lesser value). It is also of doubtful nature as other sources from 200 AD do not mention dinosaurs or large reptiles roaming the streets of The Imperium Romanum. Additionally it is a narrative and not a relic. The motivation for writing the the text is also questionable - if dinosaurs walked every street why record it in this way? What’s the use of writing that down - please remember writing something down is quite expensive.
This is just a quick overview and by no means everything about proper handling of sources but I think you get the idea.