I get the feeling that a lot of this game must be completely different if you play as a Sith class. As a Bounty Hunter, questing in these worlds left me with the impression that the majority of Sith are completely insane. They just never seem to have problems us normal people can relate to.
Anyway, as soon as you get to the fleet you get a quest for the Black Talon, the first flashpoint. Hold that for a few minutes and head off to choose your advanced class. I think we all know what I chose: POWERTECH, THE FIRE OF COURAGE! There isn't any switching of ACs later on so make sure you get the right one. Train Flame Burst once you are a Powertech. After that, go over and get the crew skills you decided on. Note that for non-crafting professions you can start sending your companion on missions right away (if you don't need her) by bringing up the crew skills window and clicking on that profession's icon. This may seem obvious to most people but I didn't realize it until almost level 20
Now you can finally run your first flashpoint! This flashpoint can easily be done with two people and two companions. It is designed to not really need a tank or healer, which is great because we don't get Ion Gas Cylinder until level 14. This place drops a lot of gear that is nice up until about level 20. It also drops a custom (orange) quality chest and blaster, so if you like the look of them you can just keep upgrading their parts and even use them into endgame if you so desire. Black Talon is short, sweet and straightforward. The only real piece of advice I have is don't bother trying to tank (no aggro without IGC). Also, when fighting Yadira Ban she will occasionally do this move where she stops attacking, swirls start appearing around her, and your movement speed is sharply cut. Walk out of melee range or you will get CC'd when her channel finishes.
PVP is available at this level as well. Click on the icon by the minimap to queue up for a warzone. Don't worry about being low level. What happens in this game is that after the players for each side are chosen, the lower level players in the warzone have their stats artificially boosted to be on par with those of the highest level player. It's not a perfect system but it is a significant improvement over some OTHER games where you effectively can only PVP every five levels because of stat differences in your bracket.
On Dromund Kaas, I decided to just start skipping the heroic quests because I was having trouble assembling groups. I completed every single player quest though, finishing at about level 18 with a couple Black Talon runs. It is quite a depressing planet, what with the overcast sky, large jungle animals, and crazy Sith everywhere (Mako feels my pain). Can't say I was sad to see it go. Plus, after Dromund Kaas you get your first starship (and second companion) so who could be sad about that? Once you have 2V-R8, always be sending the companion who isn't with you on crew skills missions. Your wallet will thank you later.