Sue enters the book kind of late in the game, in Part Second, but once she shows up, Sue grabs the audience (and Jude) and propels the novel forward.
While Jude gets to know Sue first through her portrait in his aunt's house, she really becomes a character in the book when she seeks Jude out in Christminster.
They hit it off, but Sue tells Jude she has to leave soon.
To keep Sue close by, Jude helps Sue get a job working with Jude's old schoolmaster Phillotson as a teacher. This keeps her nearby, but it proves to be a bit of a sticky situation—after all, Jude and Sue are doing this weird courting/not courting/hey-we're-just-cousins thing even as Sue and Phillotson begin working together.
With the help of Phillotson, Sue attends a teaching school in Melchester.
Romance alert! Once Jude moves nearby, Jude meets up with Sue, and they start spending a lot of time together. They can say they're just friends or cousins or whatever, but there is clearly something going on between the two of them—and unfortunately, we're not the only ones who see the sparks between the two of them.
Scandal alert! Sue stays out all night with Jude, which causes a scandal at the school.
Not wanting to deal with solitary confinement as a punishment for her "offense," Sue escapes the school and goes to stay with Jude.
This is an especially big deal because Sue is engaged to Phillotson.
Here's some advice: if you have a girl you really, really like, you probably shouldn't hide the fact that you are actually married to another woman from her. Sue learns that Jude was married to Arabella. Her response is, shall we say, not calm.
Sue marries Phillotson.
And hey, guess what, Sue is unhappy in marriage to the guy she finds dull and kind of repulsive. Surprise!
After the situation gets too horrible for her to bear, Sue leaves Phillotson and goes to stay with Jude. Even though they are living together, they do not have sex or engage in any kind of physical relationship—Sue just isn't into the whole idea.
Eventually, she divorces Phillotson and Jude divorces Arabella.
To stop Jude from seeing Arabella when she turns up once more, Sue throws herself physically at Jude. From this point on, the relationship changes.
Deeply in love and seemingly very happy, the two go through life together but unmarried.
(Of course, they may be happy, but the people around them are furious that Jude and Sue are just ignoring how things are supposed to go—their reputations are not great by this point.)
Arabella sends Jude their son, Little Father Time, and Sue does her best to raise him as her own.
Sue and Jude have two of their own children.
Once again, here comes the biggest shock in the novel: Little Father Time kills the babies and himself. Sue and Jude find the bodies.
The grief causes Sue to have a miscarriage.
Deeply depressed, Sue turns to religion, which leads Sue to leave Jude and remarry Phillotson.
At their final meeting, Sue kisses Jude passionately and professes her love to him. However, she can't let go of the guilt she feels for living such an unconventional, unreligious life.
Sue leaves Jude forever, even though she knows that he is ill and suffering.
Sue has sex with Phillotson, thus sealing their marriage—even though she feels physically sick at the idea.