My advice is to build on what you know. As you do this more and more, your skills will expand naturally.
As you read, when you come across something that rings a bell, either within the work or outside it, jot it down in the margin. This will require you to emerge from the flow of the book momentarily, and at first you may find it annoying to do so. But as you get used to it, the process will become less intrusive. Once you discover the joys of it, it will actually become part of the thrill of reading.
Do not highlight or underline the text itself. When you go back to re-read, your focus and interests will have shifted, and the mark-up will be distracting. You don’t want your books to look like Cpl. Klinger’s paperbacks with all the naughty bits underlined in red.
The “cf” in my “cf 67” example above indicates “compare” or “see also”. This is a useful tool to link to other pages within a book. If you start noticing themes or images recurring, start noting them. Then, as you progress, when you see more recurrences, add “cf” notes. For some highly symbol-dense books, such as Moby Dick, I even use hand-written icons, such as a small flame, at the top of the page for particularly important motifs.
Watch for images that get a lot of space, more than they seem to deserve. For instance, Melville spends a little ink describing the lantern that stays upright even as the ship lists (being suspended from an arm with a pivot). This image of the upright (righteous) flame is indicative of a certain moral and spiritual perspective that runs throughout his work, not just this novel.
When part of a story reminds you of other stories or actual events etc., note that, too. It doesn’t matter what it is – could be a psalm or a nursery rhyme. If you’re reading Faulkner, you’ll be making a lot of Biblical notes. If you’re reading Pynchon, you’ll be looking at everything from modern physics to cartoons to advertising slogans.
If you find a passage strikingly beautiful or insightful or poignant, make a note. Chances are, you’ll want to find it again, and your notes will help. I often simply put a ! by passages of this sort.
Look for wordplay. Hardly an author worth his/her salt doesn’t enjoy wordplay. For example, in Faulkner’s Absalom! Absalom!, Quentin’s roommate is a fellow named Shreve. Very close to shrive/shriven – to confess one’s sins, to be freed from guilt, to be administered the sacrament of reconciliation. And indeed, Quentin uses Shreve as his sounding board as he plumbs the meaning of his experiences. Shreve is a sort of confessor – although one who does calestenics while half-naked in front of an open window in winter.
Speaking of Quentin, some books practically demand that you keep notes. Faulker especially, can be difficult to follow without notes. When more than one character shares a name (Faulkner’s Quentin Compson, Garcia Marquez’s Aureliano Buendia) you gotta keep on your toes.
Look for literary cross-references.
In The Great Gatsby, Fitzgerald locates the action in “East Egg” and “West Egg”. One can’t help but be reminded of the Big-endian/Little-endian dispute in Swift’s Gulliver’s Travels, in which a protracted war is fought over which end of a boiled egg to break first. Gives you some idea of the author’s opinion of the shallowness of that culture. But Gatsby doesn’t see it. He spends all his energy trying to become a part of that culture, simply to impress a woman he knows very little about.
In the end, when his father shows up with some of his son’s personal effects, we see parallels with the autobiography of Ben Franklin – all the rigorous self-help that supposedly should advance a man in his standing in the world. Fitzgerald is essentially putting the lie to Franklin’s ideal – no matter what he did, Gatsby could never really raise himself socially in 20th century America, at least not to that level. One must be born into that class. He dies in a doomed attempt to enter a world that’s not worth entering in the first place, and that’s part of the tragedy.
I could go on, but this post is long enough. I’ll respond to questions if you have any. Main thing – progress at your own pace, work from your own experience.
Read the important sources. If you haven’t read the Bible, read it. You can’t get all the juice from lit in English otherwise. At least read Genesis, Exodus, Job, Psalms, Proverbs, Song of Songs, the 4 Gospels + Acts. Shakespeare’s major plays are helpful – try to see good performances of them – as are his sonnets.