Right then. First things first - the premise of "the self".
It'd be more accurate if we say that "there is no notion of a permanent self
within Buddhism" - not really that there is no self, per se.
Rather, we should try not to become attached to the notion of a permanent "I", or "me", as such is an illusion. There is no permanent "me",
because "me" is a constantly changing concept that's completely dependent upon other factors.
It can help to look at it like this:
Each individual is made up of five factors that are subject to change. Buddha referred to these as the five khandas. These are:
Material form (the body and its constituents)
Feeling (pleasant, unpleasant, or neutral)
Perception (the operation of the senses)
Mental formations (thoughts but also decision-making)
Consciousness (our sense of being alive)
From this is becomes clear that we're not simply one idea of "self", but rather different facets to one idea, each interdependent, and each
impermanent. Thus, there is no permanent "self". Are you then still you, when you reincarnate?
Not...really. Well..yes, in as much as...the bit of you that does reincarnate still carries the same "karmic grooves" as before; all that happens
is either they get smoothed out, or added to over time. But not really, because, as stated before, there really isn't any permanent sense of
Now - because I know my limits when I'm understanding this stuff myself...let alone explaining it to someone else....I'm going to end here with a
few links that would do the job infinitely better than I'm able!
About "no self"
Bhagavad-Gita; interesting info here, too (though not strictly Buddhist in nature)
Links relating to Buddhism, karma and reincarnation
Lecture on various key Buddhist
What is reincarnation?
Essential source of Buddhist teachings
Happy reading! (though I'm sorry I can't personally add more to this - the experts do it so much better than I!)