As to why Frodo went off on his own: If you recall, the one member of the Fellowship (Boromir, the man who got killed at the end) became overcome with desire for the Ring and tried to take it from Frodo. Frodo realized that the Ring was breaking the Fellowship, and decided that the only thing he could do to prevent the Ring from overcoming them all was to leave and go off to Mordor alone (although Sam, his partner, tagged along too).
The other members of the Fellowship could only go one way or the other-- and Legolas and Gimli would follow Aragorn, since he was their leader. Aragorn realized the threat the Ring posed to the Fellowship, as Frodo had, and decided that the fate of the RIng was out of his hands. He knew that Merry and Pippin would most certainly die sooner or later in the hands of the Uruk-hai, so he and the others decided to attempt their rescue.
If you read the books later, this will become clearer, since Tolkien explains all the intertwining motives of the various characters. But what I put above is the basic idea of what’s happening.
Any other questions?