I've decided to take the lists in both Exodus and Deuteronomy to illustrate the similarities and differences between them. I'll use other verses as appropriate to explain and expand upon them. I'll be sticking with the King James translation unless otherwise specified. Although it's not always the most clear of translations, it's the one I'm the most familiar with, and I love the beauty of the language.
With that said, on to part 1. I'll be taking the first two commandments (1. No other gods, and 2. No graven images) together in this post because they're conceptually related. It's hard to mark where one ends and the other begins. (In fact, a quick glance at Wikipedia shows that Jews, Catholics, and Lutherans view these as one commandment, not two.) For this segment, there is no meaningful difference between Exodus and Deuteronomy, so I'll just use the text of Exodus.
I am the LORD thy God, which have brought thee out of the land of Egypt, out of the house of bondage.This section starts with an introduction of the speaker - God - and a reminder of what He has done for His people - freed them from slavery. Just as He freed the children of Israel from bondage in Egypt, He frees us from the bondage of sin.
Thou shalt have no other gods before me.
Thou shalt not make unto thee any graven image, or any likeness of any thing that is in heaven above, or that is in the earth beneath, or that is in the water under the earth.
Thou shalt not bow down thyself to them, nor serve them: for I the LORD thy God am a jealous God, visiting the iniquity of the fathers upon the children unto the third and fourth generation of them that hate me;
And shewing mercy unto thousands of them that love me, and keep my commandments.
At its most basic level, this is a commandment of monotheism. We are to worship God and none else. I doubt many church members have a private pantheon, so on the basic level, we're probably doing fine. However, there's much more to it than this. How many of us allow people to get between us and God? Who do we follow when the whisperings of the Spirit conflict with our ward members' express or implied cultural expectations? Are we putting money or fame or power ahead of following God? If so, those become other gods we're putting before God.
The graven images portion is less clear to me. A literal interpretation would seem to prohibit most art. It would eliminate statues, paintings, photography, etc. of people, animals, plants, and nature scenes. ("any thing that is in heaven above, or that is in the earth beneath, or that is in the water under the earth") However, I think it's tempered by the following statement "Thou shalt not bow thyself down to them, nor serve them." In context, I think it's more of a prohibition of idolatry, not a prohibition of art.
As far as the part about visiting iniquity onto the third and the fourth generation, I don't think that means God actively curses innocent children for their parents' sins. That flies in the face of justice, and God is just. (see e.g. Article of Faith 2 "We believe that man will be punished for his own sins and not for Adam's transgression.") I take this portion to be a warning that the choices of parents will have an effect on their children. If children are not raised with the knowledge of God and His truth, then they will not as easily gain the blessings of the gospel.