Aseret HaDibrot
These ten fall into two broad groups, ones governing our interactions with God and ones governing our interactions with other people. How could one choose between those groups? You can't; they're both critically important to being a committed Jew.
It is, of course, possible to be a good person without the God-centered ones, and much of what's on that latter list is stuff we all broadly agree on anyway -- don't murder, don't steal, don't bear false witness, People might haggle over adultery, coveting, and honoring parents. One approach is to say that honoring God does no good if we don't treat each other fairly ("would that they ignored Me completely but kept my torah"); another is to focus on the stuff we don't already all agree on.
I decided to re-interpret the question, in fine talmudic tradition. :-) Are some of these ten derivable from others? That would create second-order dibrot, ones you get for free based on others. Among the people-centered ones, I could argue that (at least) murder, stealing, adultery, and false witness are consequences of coveting, so the key one there is the last. (Yes, sometimes people murder not for material gain but out of jealousy or vengeance or some other emotion. But don't we, in those cases, choose to nurture those negative feelings, to hold onto them and not let them go, the way some people go out of their way to be offended or downtrodden? Isn't that a form, or at least a cousin, of coveting?)
On the God side, I think everything derives from the first -- Anochi Adonai Elohecha, I am your God (who brought you out of Egypt etc). (For my Christian readers: we divide up the commandments differently than you do; I found a helpful comparison of the different versions here.) Interestingly, when we looked at some commentaries on this, it seemed that most folks focus on the first two words -- "I am God" -- arguing that God is all-powerful and did good things for you, so he has the authority to command. (Some soften that.) I, on the other hand, gravitate immediately to the possessive Elohecha -- it's not just that there's this divine being out there, but that this is our God, the one we're in a covenant with. I'm not obligated to God because he can smite me, though that could make obedience seem like a good idea; I'm obligated to God because I accepted that obligation by affirming the covenant. Therefore, I am obligated to do my best to understand what God wants of me, in balance with all the other factors that affect my behavior (like treating other people well). Because we ascribe benevolence to God there's a good chance that treating other people well will line up with what God wants, and if it doesn't we have some hard choices, but no one ever said this would be easy.
For me, if God isn't central then there is no commander, so the rest of it boils down to independently-derived ethics and interpersonal expedience. But I don't want to stop with just the ethics I could derive on my own, important as those are; I am willing to do the stuff I don't necessarily understand, like Shabbat, because God seems to want that.

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Strictly speaking, the latter is not a translation of "Aseret HaDibrot"; the Torah actually says "Aseret HaD'varim". Apparently the Rabbis thought that was a bit silly, since none of them is a single word (although some of the later ones come close). :)
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The ban on graven images is specifically about creating objects to worship; that's why art is ok by us (but not by Muslims, who read this differently).
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