If man is to live life by his own judgement, then wouldn't people have differing moral codes since they choose different values? In other words, what one person considers to be immoral may not be considered immoral by me. Who is right and who is wrong?
There is no supreme conscious authority in Objectivism whom everyone is allegedly duty-bound (and coerced by force) to please and obey. Furthermore, Objectivism does not demand that everyone live by the same values beyond the three cardinal values of reason, purpose, and self-esteem -- and living by those three values isn't a moral commandment or edict, but a factual identification of what reality demands from man if he wants to sustain and strengthen his life qua man (meaning his existence on earth as a rational being, with no other, non-earthly mode of existence recognized by reason as attainable by man).
There is only one issue in which Objectivism endorses the use of physical force: retaliation against those who initiate (start) its use against others. Beyond the ban on initiation of physical force (enforced by government as man's agent of retaliatory physical force), Objectivist morality and Objectivist political theory leave everyone free to live (or not) as he sees fit, trading with others if he and they are willing and able.
There is no "problem" of what to do about non-Objectivists. Objectivists are free to leave them alone and to be left alone by them. If they wish to die, they are free to do so (peacefully). It's their choice (and everyone's choice). The essential moral challenge, according to Objectivism, is what to do about one's own life, given that one is free from coercion by others. One need not be concerned about others beyond respecting their individual rights (including their property) and offering to trade with them (materially and/or non-materially) if one sees value to oneself in doing so and has something of value to them to offer in return. And respecting and sometimes admiring others as fellow rational beings tends to be an important aspect of trading with them efficiently, effectively and repeatedly.
Update: Retributive Retaliation
The comments assert that Objectivism somehow uses "retaliation" (in "retaliatory physical force") differently from the usual dictionary definition, and that the usual meaning is "not the concept Objectivism actually wants." The commenter did not offer any references to any actual dictionary definition nor to passages in the Objectivist literature that indicate a different usage.
When I look up "retaliate" in a standard dictionary, I find something like "to return like for like." In the context of retaliatory physical force, this means punishing the wrongdoer, not merely stopping him in the act nor obtaining restitution (if possible) nor deterring future occurrences. Punishment of some kind (commensurate with the perpetrator's initiation of physical force) is exactly what Objectivism means by "retaliatory physical force." In the entry on "Retaliatory Force" in The Ayn Rand Lexicon, a key excerpt from Galt's Speech states:
It is only as retaliation that force may be used and only against the man who starts its use. No, I do not share his evil or sink to his concept of morality: I merely grant him his choice, destruction, the only destruction he had the right to choose: his own. He uses force to seize a value; I use it only to destroy destruction.
This is retribution, not merely attempting to stop a force wielder in the act, or to reform him, or even to deter him from similar acts in the future. It is to punish him. If I remember correctly, Ayn Rand was once asked in a Q&A session (possibly in the tape recorded lectures on Objectivism by Leonard Peikoff, before OPAR was published) if she favors "retributive justice." She replied very strongly, "Yes!" I do not see how this is different in principle from the usual, conventional meaning of "retaliate."
Ayn also reiterated this view of retaliation in a very long letter to a philosopher (dated April 29, 1961), later published in Letters of Ayn Rand, pp. 544-563. Topic no. 7 (pp. 558-560) pertains to justice, upholding the principle of punishment rather than mercy or deterrence or rehabilitation:
The law should: a. correct the consequences of the crime in regard to the victim, whenever possible (such as recovering stolen property and returning it to the owner); b. impose restraints on the criminal, such as a jail sentence, not in order to reform him, but in order to make him bear the painful consequences of his action (or their equivalent) which he inflicted on his victims; c. make the punishment proportionate to the crime in the full context of all the legally punishable crimes.
Refer also to Ayn Rand's answer to the question, "If a man infringes the rights of another, what is the moral justification of incarceration as punishment, as opposed to monetary retribution?" (Ayn Rand Answers, p. 45.) Her answer emphasizes the need for punishment, not rehabilitation.
I cannot imagine where the commenter could have acquired the idea that Objectivism doesn't endorse retributive retaliation, if that is what the commenter is referring to. The original sources show that Objectivism does endorse it.
(This view of justice has major implications for foreign policy, too.)