The Mississippi River is central to the entire book, both figuratively and literally. And many writers have used the river this way. One example is Heart of Darkness, by Joseph Conrad.
Rivers are often symbols of journeys, with good reason, since until modern times, that was often the best way to get from one place to another. Sometimes the river is a symbol of our journey through our lives.
Sometimes the river is a symbol of a hurdle we must overcome. That is where we get expressions such as "I'll cross that bridge when I come to it," and lines in old folk songs such as, "One more river to cross." In fact, that line is from an old gospel song, and has a religious meaning as well. In mythology, one went from life to death by crossing a river, and we occasionally refer to that meaning today as well.
There is a famous saying, I think from Herodotus, that one can never step in the same river twice. If you think about that for a minute, you will realize what an interesting statement that is. A river is constantly moving and constantly changing. If you dip your foot in the river one second, the next second, if you dip your foot in the river, the water is different water, isn't it? Everything has changed. This characteristic of the river makes it a symbol of constant change.
In this particular story, Huck and Jim begin their journey on the Mississippi River, so the river is really part of the plot and keeps it moving (pun intended.)
What does the river represent for Huck and Jim? The river represents freedom for each of them, though in different ways. You must remember that when this story takes place, slavery is legal in the United States and Missouri, where Huck and Jim are, is a wild and unsettled place, really almost what we would consider the "wild west" during that period. (In fact, the story of how Missouri became a state in which slavery was legal is interesting story, and I am providing you with a link about that, too.) Now, Huck wants to be part of the wild frontier and Missouri is getting too civilized for him. and he does not want to be "civilized" by anyone. So, his ride on the river is about freedom to be an uncivilized person in the Territories, the part of the country that is even less civilized than Missouri. He is almost like a wild animal that does not want to be tamed. Jim, on the other hand, is on his river journey because he can attain literal freedom from slavery if he reaches Illinois, which is a free state. So the river is a literal and figurative means of freedom for both of them. Additionally, since the river is a symbol of our journey through our lives, the river also represents Huck's journey to maturity, in spite of the fact that he does not really want to grow up.
I must add as an aside that I am pleased to see students reading Huckleberry Finn. Many school districts avoid it because it contains the "N" word, but that is really how people spoke in that time and place, and it is one of the greatest American novels ever written. I hope you are enjoying it.