moving from chapter to chapter..

How can I move my characters from Prophecies to Factions to Nightfall? Do I have to be level 20?

No, you do not have to be level 20 to cross between the first three chapters.

Look for the quests "Mhenlo's Request" and "Sunspears in Kryta" in Lion's Arch, "Chaos in Kryta" and "Sunspears in Cantha" in Kaineng Center, or "Terror in Tyria" and "Plague in Cantha" in Kamadan. These quests will allow you to move your characters between the chapters.

Note that you have to own the chapters to get to them though.

Once you do the above mentioned quests and follow ups, you can just use the "boat" to travel between them.

You do have to advance to the required parts of the chapters to connect them. You can not connect right from the start.