Specifically, right now I'm super interested in the stories surrounding Reilig Odhráin, but I'm not really sure how to find information about it other than Wikipedia and random blogs that claim to have the "real version." So for this specifically, but also folk myths in general, are there certain places I could go (or books or journals I could read) to get reliable/accurate information? Moreover, how can I tell if a source I'm looking at is to be trusted, since folklore seems to be something necessarily passed down and distorted over time?
(Hopefully this is the right sub to ask this, I'm not 100% where else to ask!)
A good place to start would be with this sub: if I were you, I would ask a question of /r/AskHistorians about Reilig Odhráin/St. Oran's Cemetery associated with the Iona Abbey, asking for any information about this place (give the locational information so that it is clear what you seek as it may garner better answers).
There are several excellent folklore journals. Britain's Folklore Society is one of the oldest of its kind in the world, and it has an excellent journal, and given your question, it may have the focus that will interest you. It can be technical, so it may be best digested after getting a few basics under your belt, but I wouldn't hesitate to peruse the back issues.
There are also some reliable books - and a lot of unreliable ones! - so I understand your trepidation, but with care and caution you can start untangling and sorting out the mass/mess of publications. Again, specific questions to this sub can inspire answers about sources that may be of value to you.
A word of caution about what you're asking, however: you ask "how can I tell if a source I'm looking at is to be trusted, since folklore seems to be something necessarily passed down and distorted over time?" People often think that there is a "true" or "original" form of a myth or a folk narrative, and all the others are "distorted" as you say. Oral narrative is by nature changeable and constantly transforming. While there are some really horrible literary retellings of folk narratives, in its oral setting (and when professionally collected) one version is as valid as the next, even if they contradict one another. It is the nature of the beast.
I hope this helps; ask questions if I can help further.
In addition to the excellent sources already posted, I would say there are around 4-5 consistently reliable places to find well-researched, well-sourced, reliable information about folklore, in order based on consistent reliability and sourced information:
While I can't help with the specific story you're looking for, as I don't personally know anything about it, I can give you some specific resources that might be useful in your search process. I've previously listed some of the main critical academic journals dealing with fairy tales and folklore here, but I'll list out a couple of the more prominent ones that you might look through:
Any one of the above journals might have something on what you're interested in and are long-standing, reputable, peer-reviewed journals. I also recommend SurlaLune and the University of Pittsburgh for finding the texts of fairy tales, folktales, and some mythological stories. Apart from these, I would try looking at the sources in the Wikipedia article; are there any books or academic papers listed? If so, try and find them on any of the academic portals like JSTOR or EBSCO Host. The Wikipedia source list is also a good resource for you in finding the book or source for the actual text, if the site has any. In general though, if you're looking for the actual text of a folk text (especially folklore or a specific variation of a fairy tale), your best bet is always going to be some type of type-specific anthology. If it's a relatively known tale or legend in any way, shape, or form, it's usually going to be in a "Celtic Tales and Myths" or a "Classical Greek Myths" or a "Scandinavian Folklore" anthology somewhere, and your biggest problem is figuring out not where the text is, but which anthology it's in or where those stories are collected/stored. When I was writing my senior thesis using Search for the Lost Husband stories (Beauty and the Beast stories are technically a sub-type of this tale type), for example, I ended up looking through a lot of BATB type-specific anthologies that specifically focused on collecting BATB tales from around the world.
As for understanding the reliability and trustworthiness of texts, it's unfortunately a life-long skill and isn't something that you can just check off a checklist and go "okay! It's definitely credible!" Essentially, you have to know what type of sources are generally reliable and learn to cultivate critical evaluation of sources you find. There are some really helpful baseline guidelines that sites like PurdueOwl and Columbia College have produced, but ultimately it's up to you to decide whether or not the author of something you're reading is a) knowledgeable enough about the topic for their words to be trusted/actually knows what they're talking about, b) is using up-to-date information, c) is well-sourced enough for their information to be reliable, d) is engaging in best research and writing practices, and e) where the source is found.
Unfortunately, this process is an art rather than a science and while there are a lot of really helpful guidelines and warning signs, there is no "one size fits all" approach. You can generally trust well-sourced academic books and articles from peer-reviewed academic journals as well as information put out under university publishers and websites, though bias and time are still problems you have to watch out for. If you're looking into the history and folklore of a specific place or town, I suggest looking up the local or regional historical society/books on that region's folk history. Organizations specifically dedicated to the collection/dissemination/study of fairy tales and folklore are also usually very credible sources. Everything after that is a guessing game and depends on how well-sourced and researched the author in question is; as you learn more about your topic, you start to understand and learn what information is generally considered credible for that area of research. I heavily suggest utilizing your local library or local university library and the librarians for help in this matter (even if you don't belong/go to the university! Librarians are usually happy to help patrons regardless).