What a difficult question! I believe (but could be wrong) that you are asking a question in terms of meaning, social significance, psychology, perhaps raising an ethical matter... There are two broad, distinct views to consider: one views individual persons as part of greater wholes --either in terms of societies, tribes, families, the state or the collective, perhaps a religious community or tradition. Another views the individual in terms that are very much anchored on a person's own values, desires, beliefs, action. So, the first is a kind of external point of view: how is the individual seen or should be seen in a larger context...while the latter is more internal. I suggest that a reasonable position would take the middle ground. An extreme internal position would seem to be close to absurdity: if I think I am a great musical, athletic egg, it is probably reasonable to think I am delusional. And an extreme external position would seem to be very dangerous. In some forms of Marxism, for example, your freedom and character are assessed in ways that seem to crush the understanding that individuals have an integrity of their own.
Philosophers have tended to think this question (who am I?) is pretty foundational, because it seems that some kind of answer is necessary in the course of addressing such questions as: what should I do with my life? What can I know about the universe or God or right and wrong? What sort of political form of government should I support? Where did I come from? Do I owe any obligations to my parents or society or the state or the religion in which I was raised? Am I a Jew, Christian, Muslim, Hindu, Buddhist, atheist, agnostic......or some combination? What can or should I hope for in life or (if there is one) a next life? One of the earliest philosophers in the west, Socrates, is said to have claimed that the unexamined life is not worth living. Whether he was right or not, it seems that without some self-examination (what do I really desire? what sort of a person am I? do I actually care about others or am I faking it?), it is very difficult to grow, to love others (if I do not know who I am, how do I know whether I love someone else?) or to dedicate your life to some great value (justice, compassion, and more).
Personally, I am on the Socratic side in this matter, but one might also worry that the question "Who am I?" is pretty individualistic, and so it might be good to balance with a question like "Who are we?" and then think who is included in the "we."
As for first steps in addressing the question of personal identity: I would recommend something Augustine (5th century) thought, and that is, your identity is partly a matter of what you love and desire. So, one thing to ask yourself is: what or whom do you love and why? The object of your love may be a person or a community or family or yourself or an ideal or God or... Augustine thought that you need to begin with some kind of self-knowledge and then find a community of fellow inquirers who welcome the practice of philosophy -- a community which genuinely seeks to love wisdom. Such a community might even suggest that the first question of the greatest importance is: What is wisdom? For a great beginning book in philosophy I recommend a classic, The Story of Philosophy by William Durrant. It is hardly ever used today, and is flawed in this or that way, but it is a narrative history of philosophy that is exciting, written with passion, and will take you on a journey through lots of different philosophical ideas about wisdom.
Good wishes, Vikram! Charles
A CLASSIC case! This is a major issue going back to ancient philosophy. The example used then was the ship of Theseus (a Greek hero). Imagine you have the ship of Theseus and a similar ship side by side. First you switch one part (the mast, say). Is the ship of Thesus still the same? Many of us want to say 'yes,' but then we get puzzled as more and more parts are switched until eventually it seems the ships have changed places. One route that philosophers have taken might bring peace to your family: some philosophers distinguish a strict sense of identity from an identity that is "popular and loose." On a strict view, you are right. Any object with parts is not the same if even a single part is removed. This is technically called mereological essentialism. According to mereological essentialism, your father's body today is not identical with the body he had as a boy. You might even suggest to him that while he went to first grade, that (pointing at his body) did not. You can retain mereological essentialism while also allowing that sometimes we can and should meaningfully speak of sameness of identity that is not strict. In the later, though, there will be different conventions that come into play. So perhaps there is a way to allow that you both may be right? If push comes to shove, however, I am on your side on this. For a defense of our view, see the book Person and Object by Roderick Chisholm.
Great question and suggestion! While some philosophers (most notably John Locke) have claimed that the key to personal identity is memory, probably the majority of philosophers today do not. Most grant that you might endure as the self-same subject despite all kinds of memory loss and replacement.... So, if it is a fact that no one does remember their past lives, it would not follow (on many accounts of what it is to be a self), this may be only a problem of epistemology and we cannot from that alone assume that reincarnation is false. Probably one reason why some today think reincarnation cannot occur is because they think that for reincarnation to occur, a person (self, subject, soul, mind) would need to switch bodies. Those of us who are dualists or who think there is something to persons more than the material body, may well grant that it is possible for a person to come to have a new body. But materialists who think that you and I are our bodies will have grave doubts about whether we can survive the destruction of our bodies.
It sounds as though you are already making a break with your past (putting "I" in quotes suggests that you think of yourself as a different person, for when you describe yourself as you are now, you are not using quotes). You mention family and friends. Insofar as you have made vows of life-long partnership as a spouse and insofar as you have not renounced the duties of family life (one's obligations as a parent or child or sibling) it seems you do have some prima facie duty to "keep in touch" with that "I" or self whose life is bound up with theirs. This might involve not just increased visits, but more electronic and alternative means of communication so that they have more of an understanding of your current situation (send photographs of your appartment, favorite places in the new city, etc). Philosophers disagree about the extent to which friendships involve obligations of that sort. Some think of friendship as a gift, others may think of it as a gift as well but believe that once given and received, there are some duties that go along with it. These may be trivial (not breaking confidences) or substantial (going to their aid at considerable cost to yourself). However you stand on friendship and duty, I suspect that even calling someone your friend requires that you are actually interested in them (at best, you love them), such that if you are no longer interested in them, there may actually be no more friendship there. There may be one-way affection (they may love you) but without reciprocation, the friendship has ended. Ceterus paribus, I would urge you not to let these frienships end too easily. You have a good in older friendships (years of fidelity, love...) that is precious.