The free tribble is likely only available for characters who were created prior to the launch of season 8 on Nov 12. So chances are you do not qualify for it. Anyway its a very minor reward, most players view it as useless so it doesn't really matter if you have it.
I second Timelord's suggestion of teaming up with a player with more experience who can show you the ropes. As for piloting the starship, just keep practicing. The starter ship is known to be a very slow and clunky ship, both in speed and turn rate. I know how frustrating it can be to pilot that ship, I just keep yelling "Come on, move your ass!" at my screen XD
The good thing is you should be level 4 or 5 by this time and can abandon the starter ship at level 10. You will get a choice of 3 different types of ships, Kirk's Constitution class cruiser, Oberth Science Vessel and Archer's "Akiraprise" Light Escort. There are 3 other ships to pick from but you'll have to pay money for them (and totally not worth the cost, so don't).
Cruisers are the beefy ships, moves slowly, has medium attacks but can take a lot of damage. Science vessels are faster than cruisers, has weak attacks but is good at sciencey effects that can take down shields, hold ships in place, etc.. Escorts are zippy and fast, has strong attacks but blow up easily.