We don't know Bathsheba's name the first time we see her in this book, but we do know that she's very fond of her own appearance. When Gabriel Oak watches her from across a field, the first thing he notices is that "a small swing looking-glass was disclosed [in her hand], in which she proceeded to survey herself attentively. She parted her lips, and smiled" (1.13). Or in other words, Bathsheba likes to look at herself in mirrors when she doesn't think anyone is watching. It's like checking yourself out in a window's reflection before realizing that there's someone on the other side of the window looking right at you. Awkward.
The thing that really gets Oak about Bathsheba's vanity is the sheer excess of it. It's one thing to look in a mirror to straighten your appearance, but he realizes right away that "[t]here was no necessity whatever for her looking in the glass. She did not adjust her hat, or pat her hair, or press a dimple into shape […]" (1.15). This vanity is something that continues to pop up later in the book, eventually leading Bathsheba to curse God for giving her such a beautiful face.
Sheesh. Get over yourself.
Even though she's vain, Bathsheba is still a respectable character in this novel, mostly because of the way she is able to be so independent as a woman living in 19th-century England. We can see signs of this independence early on when Bathsheba decides to ride her horse like a man, with one leg dangling over either side of the saddle. To her aunt (and to most Victorians), this is a sexually scandalous thing to do. But Bathsheba confidently responds to any objections by saying, "I can ride on the other [saddle]: trust me" (2.31).
Bathsheba really shows what she's made of when she decides to fire her farm's manager and to run the place herself. For the men working under her, this is an unthinkable thing to do. But again, Bathsheba is ready for the challenge, telling them, "I have two matters to speak of. The first is that the bailiff is dismissed for thieving, and that I have formed a resolution to have no bailiff at all, but to manage everything with my own head and hands" (10.2). She goes on to do a very good job of running the farm, which makes pretty freaking ahead of her time.
Now if only she were as good at choosing husbands…
The narrator is always quick to tell us what a rational and intelligent person Bathsheba is. The problem is that whenever she has a single irrational thought, it tends to overpower her. Or as the text reads, "Many of her thoughts were perfect syllogisms; unluckily they always remained thoughts: only a few were irrational assumptions; but unfortunately they were the ones which most frequently grew into deeds" (20.5). And there seems to be no deed that comes back to bite her more than marrying Sergeant Troy. Being a proud and superficial person, Bathsheba is more interested in a cocky, handsome young man than she is in a solid, loving partner like Gabriel Oak. But don't worry: the book is going to teach her a lesson on this one.
When Sergeant Troy first steals a kiss from Bathsheba, she truly doesn't know what to do. She's spent so much of her life acting proud toward men and holding their hearts in the palm of her hand that she can't believe any of them would ever dare to kiss her. But the thing is that she also likes the kiss, so her confusion just collapses in on itself until she cries: "It had brought upon her a stroke resulting, as did that of Moses in Horeb, in a liquid stream—here a stream of tears. She felt like one who has sinned a great sin" (28.52).