After reading her diary until chapter 1, what impression can we make about Anne?

Questions asking for subjective answers are based upon a personal understanding and "relationship" with a text and character. Subjective answers will differ from one reader to the next. These differences between readers' perspectives lie in the fact that all readers have different points of view that they bring with them into a reading. For example, a reader's perspective can change due to gender, age, culture, society, morals, experiences, or parenting, to name a few factors. As a teacher, I tend to accept most student-presented impressions of a text, as long as there is support for why the student feels the way they do. As for a general impression that Anne makes in the opening chapter, one needs to look at a few different aspects.
First, Anne's story opens with the following line: "I hope I will be able to confide everything to you, as I have never been able to confide in anyone, and I hope you will be a great source of comfort and support." Here, Anne shows her trust in us as readers and human beings. Although she does not know us, she is willing to trust us. One can either become endeared to her or estranged. The endearment would exist because she immediately trusts us and knows nothing about us. She is unwilling to pass judgment on us. One may, on the other hand, find this immediate trust alienating. One could ask himself or herself how Anne could trust someone so immediately. He or she may believe Anne to be naive and ignorant regarding the ways of the world. This said, one must also remember that Anne's "you" is her diary and not the reader. Upon remembering or realizing this, one's initial thoughts on Anne may (should) change.
Based upon the ideas mentioned earlier about reader perspective, one may feel as though Anne is quite spoiled. She receives many presents for her birthday. A reader from an impoverished background may feel like Anne is bragging at this point. Other readers may feel like Anne does not get much, if they are richer and get more than what she describes for their own birthdays.
As Anne details her "friends" and schoolmates, she uses language that many readers may relate to. As social beings, we seek out relationships. As these relationships grow, we learn about who the people really are. Anne's detailing of these relationships shows her as a strong young girl with very specific thoughts on the people around her. All ages of readers should be able to relate to her descriptions.
One statement which may make readers feel for Anne is the following: "I don't have a friend." Given our social nature, people need friendships and relationships. The idea that Anne has no friends and is "speaking" to her journal may make many feel sorry for her (especially given that most people know the story the diary tells).
In the end, Anne comes across as an honest young girl who wants nothing more than a friend.