There should be alot of quest givers in the fleet for the beginnings of the daily quest chains. I know that there is a terminal that begins your Corellia ones, as well as a Bounty Board for the Belsavis ones, and an NPC for the Ilum ones. Don't know if you have to talk with these things before you start the quests, but I would just to be sure.
Just know that the Corellia ones are strictly dailies, the Belsavis ones are bonus quests as well as repeatable dailies, and the Ilum quests are world quests along with daily ones, without any class content.
This is true, but OP will be on a wild goose chase for weeks with meaningless lowbee quests as TOR has deemed it necessary, in their infinite wisdom to make little gold triangles visible forever with no way to turn off the ones you have out leveled, much less the ones you don't want to bother with.