The discipline of textual criticism ("TC") isn't a liberal conspiracy invented in the 19th or 20th century. It's a vital aspect of Christian scholarship going back centuries with better techniques being developed in the 20th century. It's the apologetics of the New Testament text. We need to understand how proper scholarship protects the New Testament from quaint, apocryphal verses slipping in.
The basic principles used in TC are neither esoteric nor reserved for academia, but are basic enough for anyone to grasp. Engineers and scientists will feel at home with the principles as they're no different than what's used to identify and study anything, but the fundamentals are within reach for everyone. Ultimately, TC is about the means God uses to protect His text (cf. WCF 5.2,3).
TC will help you understand John 7:53-8:11 is exactly as canonical as the menu at Chipotle. It was likely the result of a scribe writing a quaint Max Lucado-like story and storing it inside the New Testament he was currently copying. The next scribe who came along thought the previous scribe accidentally missed a page and included it as an insert (which happened often), so he included it in his next copy. That's why it shows up at Luke 21:3 too: the bookmark was there when another scribe was copying. A moving story is one thing, a story that moves around is quite another. The fact that is also shows up at the end of John at John 21:25 and the end of Luke 24:53 should put away any doubt in our minds that this is a devotional story, not a canonical part of John.
Imagine a new edition of Lord of the Rings: The Fellowship of the Ring being released with an excerpt about how Frodo visited Hogwarts -- clearly something is wrong here, but further imagine that people get offended when you removed that obviously apocryphal section. That's how ridiculous the inclusion of John 7:53-8:11 is. Blacking this section out of your Bible is the litmus test of a person paying attention. Nobody is actually confused about this -- it's not part of the Bible. It's our job to understand why.
One issue that arises in regards to the acceptance of textual criticism stems from an unhealthy conflation of what's in the domain of mechanics or means and what's in the domain of faith. This is a problem that shows up in history on a few important places. John Calvin and Francis Turretin were not trained in textual matters, just as they weren't trained in any medical profession. So, their faith-focused view of textual data self-binds them to an improper conclusion. For what's probably the clearest analogy to this, we can read Philip Schaff remind us of the Copernican revolution:
Even after the triumph of the Copernican system in the scientific world, there were respectable theologians, like John Owen and John Wesley, who found it inconsistent with their theory of inspiration, and rejected it as a delusive and arbitrary hypothesis tending towards infidelity. “E pur si muove,” the earth does move for all that!
Misunderstanding textual or astronomical data doesn't force us to trash the works of Turretin or Owen, and neither do their profoundly helpful theological volumes cause us to accept their untrained opinions on matters better left to others. This doesn't imply that we should further our contemporary tendency to overly focus specialization by leaving each topic to the experts in that field, on the contrary we can all understand that the textual evidences and principles just as we can all understand that the Earth revolves around the sun. We must simply understand that not very topic is within the domain of the internal data of the Bible. Schaff continues:
There can be no contradiction between the Bible and science; for the Bible is not a book of astronomy or geology or science; but a book of religion, teaching the relation of the world and man to God; and when it touches upon the heavenly bodies, it uses the phenomenal popular language without pronouncing judgment for or against any scientific theory.
TC is intimately connected with the defense of revelation. You can't talk about the importance of the inerrancy without taking TC seriously. They're linked. Because of this, works on the reliability of the New Testament are usually TC materials.
New Testament Text and Translation Commentary (Philip Comfort)
You need the following resource handy. This book will give you the 90/10 of what you'd need. You can quickly lookup passage to see a summary of the textual data regarding a passage and an explanation of what the data means. You can get the raw data from a Greek textual apparatus, but this helps you understand the data without throwing your hands up in confusion while muttering some nonsense about scholars trying to change God's word.
But, just as you need to understand the terms "alternator", "drive shaft", and "blinker fluid" when talking to your mechanic (or jester brother-in-law), you need an overview of the terms and principles to understand the TC data. The Comfort text will give you an overview of this, but while I find the examples in the intro to be great, I find the intro as a whole to be too quick. You'll want to watch a few solid YouTube videos or read another book to understand the techniques better.
New English Translation with Notes
This is the other resource that's required for serious day-to-day study. "Come for the Notes—Stay for the Translation" is written on their website. It's a great translation, but you come for the notes. The notes are not meant to explain the theology or history, they're meant to explain the textual data of that passage. I've used this translation since their 1st edition was published. The 2nd edition came out in 2019. You want the full notes edition.
Sometimes you'll hear that translators translate and interpreters interpret. The only people who repeat are monolingual English-speakers. Translation requires context, the use of context is called interpretation. There is no translation without interpretation. You might translate Latin "haec magistra est" as "This is the teacher", but the moment you get to the next sentence "haec non magister est", you realized that your first sentence needed to read "this is a female teacher" since the second is "this is not a male teacher". The same works across paragraphs and entire books of the Bible.
The NET dynamic translation is beautiful and helpful for your study. You should always be using multiple translations in your study (in as many language as you can handle -- per this advice from Clair Davis). Add this to your day-to-day usage.
The Reliability of the New Testament Text (James White)
This video will fly you through some of the basic concepts and prepare you for deeper study.
The Basics of New Testament Textual Criticism (Dan Wallace)
When you see a professor, you should ask yourself: what's his real job? Normally professors teach in order to get funding for their true passion. In the case of Dan Wallace, in addition to have written one of the standard Greek textbooks, his real job is textual criticism and manuscript discovery. You can safely read, listen to, and watch everything he says on these topics. He's pretty good at explaining things.
The King James Only Controversy: Can You Trust Modern Translations? (James White)
This book has a title made for marketing (he has a "Controversy" series). It's an introductory book on TC. It will fly you through most things you need to read TC data.
James White has also has an Internet broadcast going back many decades covering these details in depth -- now he walks you through the Greek data on Youtube. It's great. I used to be an active participant, and I'm the "fella" he mentioned in this 2008 video:
Encountering the Manuscripts: An Introduction to New Testament Paleography and Textual Criticism (Philip Comfort)
This is a deeper resource for people who want to go to the next level.
Manuscripts of the Greek Bible: An Introduction to Palaeography (Bruce Metzger)
This work is not as broad, but it's what most of us read before Comfort's work was published. It's also often unholy expensive. The author was the professor of liberal critic Bart Ehrman -- an apple which fell very far from the tree. Philip Comfort is a more faithful successor to Metzger.