You do have to make a new character to play the tutorial - just like with the Federation. You can do that on Tribble if you've got no character slots.
Other missions are available from the Episodes Tab (just like the first FED mission is the Azura Rescue). They even give you tutorial messages, even if you're lvl 50. For some reason though these missions cannot be skipped and you have to complete them in order.
Also, you should play the tutorial if only to know the context behind those new missions.