I just started playing this for the first time today. I played up to the stop at Earth Spacedock after fighting the Borg at Vega, and so far I am having a blast. I noticed as I was looking through the episodes/missions that it had all of the different storylines or seasons or whatever they are, so does that mean I can jump past the Klingon War stuff and go right to the Delta Quadrant?
EDIT: Just thought of one more question, when do I get to change ships or do I stick with a Miranda class the whole time?
Welcome to STO

Barring a few compulsory checkpoint missions, you can pretty much play in any order you like. Just bare in mind most of the mission arcs are now highly serialized. At least for your first play thru, you may want to play them in order. This is especially true for the latest 3 mission arcs, Solanae Dyson Sphere, The Delta Quadrant and The Iconian War.
You get a choice of 3 ships to upgrade to every 10 levels till level 40. At level 10, you choices are Constitution, Saber, Nova. Level 20 ship classes are the less well-known Cheyenne (Picard's Stargazer), Akira and Olympic (sort of like a century old version of Beverly Crusher's Pasteur in All Good Things). Level 30 are the hero ship classes, the Galaxy, Defiant and Intrepid. And finally at level 40 and the final tier of free ships, you get the Sovereign, Prometheus and Luna (Riker's Titan).
Beyond level 40, your choices for free ships become a bit more limited. There used to be a mission, Temporal Ambassador, that gives you a free Ambassador class (Enterprise-C), but looking now I can't seem to find the mission in any of the story arcs. The summer and winter events also give out a very good free ship in exchange for playing their respective events for 14 days. There are also a collection of mirror universe ships sold cheaply on the Exchange.
Of course if you're willing to pay real world money or can gather enough dilithium in-game, then the C-Store has a huge variety of Tier 5 and Tier 6 ships including the Vesta.
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