I just read your first chapter. As I'm sure you know, the concept of starting out a fantasy story along these general lines, to establish that a person of royal blood is now hiding in obscurity, somewhere far away from home, has been done many times before. (But there are only so many basic ways to start a fantasy story, I suppose.) I admit that making the heroine the daughter of a dragon and a vampire is something new in my experience, although I get the impression that most of the time it is not obvious to the casual observer that Pixy has such an odd family tree, instead of appearing to have the shape of an ordinary human girl. I could be wrong about that, though -- I was confused when she slashed with her "claws" at the villain; I was wondering if she always had claws (instead of, say, ordinary fingernails?), or had she grown them in a hurry when she got angry at him, or what? Is there anything about her physical appearance (claws, fangs, that sort of thing) at the age of 19 that would alert people that she probably had some dragons and/or vampires in her ancestry? At this stage, I simply don't know for sure!

It's also a bit confusing the way you have her wake up from this dream-flashback of what happened ten years ago, and then at the very end of the chapter she is thinking again of what happened (I think) right after she arrived in a different world ten years ago, and then the chapter ends. I'm not sure how I would change that if it were my story, but I think I'd do some rearranging so we don't have that one short paragraph "in the present" jammed in between scenes from "ten years ago." It just doesn't seem to flow right in its current form; not to my eye, at least. Feels too much like jumping back and forth, indecisively. (But what do I know?)

Beyond that, I'll just indulge in a little Shameless Nitpicking about a few samples of rough spots that caught my eye in this first chapter; things that could stand a little more polishing.

* They ran in a hurry. The caste was burning and they needed to make their escape. Chase, the youngest of the four, tripped and injured his leg. The seven year old cried out holding his ankle and was quickly picked up by the eldest, Damien. The twelve year old preteen called out to his other two siblings, Seth and Pixy, to ensure that the nine year old twins stayed close behind. Seth held tightly onto Pixy’s hand, making sure that she didn’t fall behind. They couldn’t afford to fall back. *

A few nitpicks occurred to me immediately:

1. I think you mean "the castle" was burning instead of "the caste."

2. Calling Damien "the twelve year old preteen" is redundant. If he's twelve years old, then he is not yet a teenager, therefore we automatically know he is a preteen.

3. The way I usually see people's ages punctuated is "twelve-year-old," "seven-year-old," and so forth -- but frankly, I don't KNOW if an editor would absolutely require you do it that way. I think it's clearer that way, and probably what readers will be accustomed to seeing, but I can't swear that your curernt approach is dead wrong. This nitpick is really just a suggestion.

And down toward the end of this chapter, you say:

* She sprang up from her bed in a cold sweat, heart was pounding and her breathing matched her hearts’ quick pace. *

Saying "hearts' quick pace" is only correct if she has two or more hearts. If she only has one, then it's "heart's quick pace." (I think earlier in that sentence it should also say "her heart was pounding" or else just "heart pounding," but I admit that might just be a mater of taste.)

There are various other little things in this chapter that could stand some extra polishing, but I don't want to bore you to death with a full list. I'm just trying to illustrate the idea that there's still room for improvement.

hey! Check this out, first review!
Well first of all, let me tell you that after reading this, it took me a while to close my mouth. And I mean it in a good way.
A Vampire- Dragon hybrid. Wow. Just wow. That would be SUCH a good picture. Of course, if you send me a more detailed description of Pixy and if you allow me to draw it, of course. I usualy like to draw things that interest me and this is definitly interesting. So, keep on writing (I am SO watching you) and let me know if you are letting me draw Pixy. I don't want it to end up as stealing your idea.
Keep up the good work! ;)
~Maha~