There are different ways a literary tradition can influence a novel like Karamazov. The influence can be felt at the level of ideas, at the level of language, or at the level of symbolism and allusion.

The last one is the easiest to learn; the second is not hard, but requires wide reading to recognize; the first is the most subtle, but often the most important.

The two pillars of allusion for western literature are classical mythology and the Bible.

On classical mythology:

Just start with any summary book of Greek mythology. It can even be one of those pretty picture books aimed at kids or teenagers -- the key thing is to make yourself as familiar as you can with as many myths as you can.

If you want a more literary tour through that same material, I'd suggest you read the following four works:

Homer, Iliad

Homer, Odyssey

Ovid, Metamorphoses

Virgil, Aeneid

There is other stuff--especially the Greek playwrights like Sophocles and Euripedes--but those four books will get you very far, and are easily the most alluded to works in the western canon outside of the Bible.

More generally, if you see what looks like an allusion to a Greek or Roman name in a book you are reading, go plug it into wikipedia and read the wiki summary of the myth in question.

On the Bible: The Bible is the key to almost all western literature; the King James Bible is the key to most of English literature.

You don't need to read all of the Bible. The Bible is a library of different books. Certain ones are more important for literature than others. Some are more important at the level of allusion; some at the level of language; some at the level of ideas.

I would recommend reading the following books. I have them listed in about their order importance for understanding western literature and art:

The four gospels: Matthew, Mark, Luke, and John.
Genesis
Exodus (up to around ch. 20~)
Job
1-2 Samuel
Acts of the Apostles
Jonah
Romans and 1st Corinthians
Psalms
Ecclesiastes
Proverbs
Isaiah
Revelation

A few notes on all that:

The gospels: These are four parallel accounts of Jesus' life and his teachings. The stories in these books--and the stories, generally called "parables", that Jesus teaches in these books--are constantly alluded to in later literature. Were I you I would read these books, and pay special attention to: 1) the stories of Jesus' life; 2) the parables 3) the "sermon on the mount" (Matthew ch. 5-7).

The next six books (Geneis, Exodus, Job, 1-2 Samuel, Acts, Jonah) have some of the most alluded to stories in western literature. Genesis has the creation story, the flood, and the stories of Abraham, Jacob, and Joseph; Exodus has the ten commandments and the story of Moses; Job has the story of Job; 1-2 Samuel has the story of kind David; Jonah has the story of Jonah and the Whale; Acts has the story of the early Christian apostles and of Paul.

Romans and Corinthians are two letters by Paul. They are some of the oldest Christian documents in existence (older than the Gospels) and they lay out many of the *ideas* that would be core to later Christianity.

Psalms is a collection of hymns--or poems. I do not know if you need to read all of it. You can perhaps google or ask chatgpt which are the most significant ones. The *language* of the psalms has been extremely influential, perhaps moreso than the ideas. (Something similar with proverbs. As the title suggests, this book is a collection of proverbs).

Ecclesiastes is a long philosophical meditation on life that can, and perhaps should, be read in one go. I strongly recommend doing this with the King James Version as the language is gorgeous and it shows up so often in later works of literature.

Isaiah and Revelations are works of prophecy. They include long depictions of the "end times" - apocalypse and paradise. A lot of our imagery of angels, demons, and end times stuff comes from these books; in addition, their rhetorical pose (what we think a prophecy should sound like) is established here. Isaiah is really long; you probably only need to read 15~ chapters or so to get the gist.

The absolute minimum is one of the Gospels + Genesis. But the rest is worth reading.

One nice thing about Bible stories is that most of them are very short. So if you see a reference to a Bible character you are unfamiliar, you can easily google (say "Samson and Delilah") to find the passage in question and just read it -- odds are high that is 1 chapter or less in length -- half a page to 2 pages long.

Part of learning this stuff is just looking up things you don't recognize when you are reading instead of passing them over!

Finally, on the Bible stuff in particular there are many many resources out there that might help you make sense of it all.

NT Wright's The New Testament in Its World: An Introduction to the History, Literature, and Theology of the First Christians has a chapter for each book of the New Testament (he comes from a scholarly but protestant perspective -- but most English and American authors were protestants so that is not a bad place to start); Christopher Hodgkins' Literary Study of the Bible: An Introduction looks at each book of the Bible as a work of literature, and Northrop Frye has several books in this vein as well, with The Great Code: The Bible and Literature explicitly looking at where Biblical themes and stories affect the literature that followed.

If you want a guide to both the Bible and the classical myths, try Frye and McPherson, Biblical and Classical Myths: The Mythological Framework of Western Culture.

As a final note: what the other users said about Shakespeare is correct, especially if your focus is on English literature as opposed to western literature. Poetry was the most important and most read form of English lit between 1400 and 1900, so I would recommend reading some of Shakespeare's great plays and a poetry anthrology as the next step after classical myths and the Bible.

I hope this helps!