Generally goes from life to life is the subtle mind part of the mind (i.e. the storehouse consciousness or alaya vijnana or bhavanga citta or other terms depending on the school), but it is not a permanent, unitary, independent self. Rather, it is an afflicted consciousness.
As the Dalai Lama XIV puts it:
A: ...If one understands the term "soul" as a continuum of individuality from moment to moment, from lifetime to lifetime, then one can say that Buddhism also accepts a concept of soul; there is a kind of continuum of consciousness. From that point of view, the debate on whether or not there is a soul becomes strictly semantic. However, in the Buddhist doctrine of selflessness, or "no soul" theory, the understanding is that there is no eternal, unchanging, abiding, permanent self called "soul." That is what is being denied in Buddhism.
from Healing Anger: The Power of Patience from a Buddhist Perspective by His Holiness the Dalai Lama.