I agree with all of the above.
* Slow (way) down
* Stay where you are
* Date casually
* Get to know each other very very very well before making any sort of move to change your address.
In order to help you maintain a center, please read this:
I think we are ready to live together!!
We frequently hear about people living together who later find that one, the other, or both are not happy living together. Similarly, we frequently read a post in which a couple is contemplating moving in together and looking for a suitable residence. Here is an initial Check List.
As for her child(ren), many relationship experts recommend dating the parent without meeting her/his offspring. The reason is to keep the initial and early relationship strictly between the adults for many months. It is very bad on children to have men coming and going out of their parent's life. Until you have a successful history with this woman over the course of eight months or so, and sufficient to know if the two of you are going to continue, do not become involved in the child's life, except for a casual introduction.
The one thing not to do is to become a part of the child's life, a pseudo parent figure, only to break up with the mother/father and have the cycle repeat over and over. This can be really bad for the child(ren) no matter how good and honorable your intentions are.
Ideally, this mother should not marry until after her child has graduated high school. Her focus for the next many years must be her children. Can she day? Of course. Marry? Blended relationships are difficult; step parents are problematical the older the child is.
So, ten days is far too early to make any decisions. Carry the relationship out nine months before making any tentative plans. We understand you are enamored with her, in lust, in like, and want to have this budding relationship have a favorable journey, yet there are relationship stages that you must go through as you travel.
Have you been reading the articles listed in the Index? Has she? If so, have you discussed what you have learned?