I presume we are talking about the actions of a girl in a car, or in some other type of vehicle.
In my opinion, in neither case is it possible to be quite sure of the meaning.
"to pull away" probably means "to drive off/away" ("allontanarsi [in macchina]"?), but it could also means "to drive faster and so to open up a lead".
It isn't possible from the context given to say with any certainty what "to pull off to the side" means. It could mean "to pull over", in the sense of "accostare", or it could mean "to swerve" (deviare/sbandare).