Emma Woodhouse is about 21 years old, lives with a father and a governess but no longer has a mother. Because of her wealthy situation and her upbringing where the adults did not discipline her much, she's been almost a brat.
I wouldn’t call her a “brat”, but yes, due to the fact that she has no mother to admonish or reprimand her when she has done something wrong, she has gotten used to getting things her way. She is described as “clever”, so clearly, she is able to think for herself and is able to clearly articulate herself to get what she wants (and this is confirmed in third paragraph.) Also, in the book, the author pointed out that the governess, Miss Taylor, was “less a governess than a friend”, which lead to them more like “sisters”. In addition, this sentence clearly explains it all: in reference to the governess: “the mildness of her temper had hardly allowed her to impose any restraint”. This is a direct way of saying that the governess was gentle and mild and did not dole out any punishments or set any “rules” to be followed as in a teacher/student relationship normally would. This gave Emma every opportunity to do whatever she pleased, and, although Emma had high regard for her governess’s judgments, she was clearly the “mastermind” of all their activities thus far.
having too much her own way and thinking a little too highly of herself. So says the author. The author sees this as a danger, although it's not clear what that really means. At least not at this point.
I’m going to say that the author really was quite clear. It’s called foreshadow, and you know what it means so I won't go into it. Emma is wealthy and secured in her lot in life, as you’ve said, so with this security and stability, the author said that she possesses “the power of having rather too much her own way, and a disposition to think a little well of herself” --- THIS is the foreshadowing of the many conflicts to come because Emma is too stubborn (because she always gets what she wants) and too self-assured to the point of snobbery (as someone said) and too vain in her self-assuranced all due to her wealthy station in life.
--In Chapter 1, we see some descriptions of Emma's backgrounds, her father's situation, Mrs. Weston's marriage and a few others. Mrs. Weston's wedding took place in this chapter. Mr. Knightley appears near the end of the chapter. Mr. Knightley and Emma seem acquainted and had small friendly argument with each other. A friendly argument? How can an argument be friendly? You may ask. Well, that's just my say. You say it how you want it. 
Ok, you said we can say it however we want and I don’t have a term or a phrase, call it a sparring of wits, but I would have to say that Emma and Mr. Knightly, as described in the book, are very familiar with each other and are at ease with each other, so they can joke about and point out each other flaws to one another without being offended. Their rapport is easily seen and felt to be free of any evil or mean intentions. So, when I read this book for the first time, I already knew that Mr. Knightly was going to be voice of reason throughout the book. And he demonstrated it when he asked Emma if she really was “successful” because she actually planned and strategized to get Miss Taylor and Mr. Weston together or was it merely a “lucky guess”. And I’m sure we can all relate to this notion. Sometimes, when something happens, we like to think that we already saw it coming, when, of course, we never went out of way to do anything to make it happen. Know what I mean? LOL
The chapter seems very concerned about the detachment Emma and her father feels in Mrs. Weston's moving out of their house. Not sure what meaning that has for the novel. But the author makes a big issue out of this.
I think the author made a “big issue” of this because she wanted to set up the upcoming storyline. We were told that Miss Taylor had been with them for quite a long time and she was practically a part of the family. She was the buffer between father and daughter, plus, she was able to keep everyone lightly entertained because she wasn’t really of blood relation. Does that make sense? So she was able to balance life with the father by being the proper governess and adding some sense of authority (even though we were clearly told that she’s too mild-temper to be of any consesquence) and mother-figure to his daughter; yet at the same time, she was able to be a confidant, a friend, a constant companion that could match Emma in wit. And to add to latter point, Emma profoundly misses Miss Taylor because of the fact that her father was not able to articulate and match her wit to wit. She was “clever” so she needed and depended on Miss Taylor’s equally “clever” personality to keep the days and nights going. And to me, the author made a “big issue” over this because this is actually where all the conflicts begins to take shape. Here, in Chapter 1, we are getting a sense of what is to come in later chapters. Emma is clearly and plainly BORED OUTTA HER MIND and because she thinks “too highly of herself”, she thinks she was the cause of the marriage between Miss Taylor and Mr. Weston, so being detached and voided of anything to keep them occupied, they – er, I mean, Emma, now has all the time in the world to take on this skill that she thinks she has – matchmaking! And that is where all the upcoming plots begins….
