When I was young I didn't understand why. In fact it seems the were headed to New Orleans. In both cases it would be because this was during the war, and he was trying to meet the Union army, where the union had already proclaimed that slaves of the states in rebellion were free.
As for Huckleberry's part, he was taken by his father, and locked in his cabin. He managed to escape, and was also headed down river. Jim fashions a raft, and together the head down river. They have many adventures, mostly with Jim staying with the raft, while Huck goes to explore. Huckleberry ends up in the middle of a feud, and barely escapes with his life. He and Jim meet a couple con men (Harvey Korman and David Wayne), and Huckleberry and Jim barely escape with their lives when the pretend to put on a show, but then make off with the ticket sales with no show. Huckleberry then ends up with another family. The con men pretend to be long lost relatives of a deceased person, and want to claim the inheritance. Huck swindles them, and puts the money in the casket. In the meantime the rightful relatives show up, and things are bad for the con men and Huck says where the money is.
However when he returns to the raft, Jim is gone. The con men turned him into the slave hunters, and he is headed back to slavery, instead of to Cairo. Huck manages to set him free, and they run away chased by the coon dogs and men. Huck trades him shirts, agreeing to go back to his life with the old women, while Jim escapes. The movie ends with Jim getting away while Huck watches.
I liked the way this movie presented the desire to be free, and the sacrifice for freedom and family. It was also very true to the book. The music was forgettable.