Well things could be changing with the addition of PvE content for us but in the past the best way to get experience is to do PvP (both Ground and Space).
This is due to the fact that we get many missions that can overlap. Such as kill another player captain, destroy x number of turrets, die to another captain and so on. So in doing one PvP match you could be satisfying more than one quest.
As for the best maps I would stick to the capture and hold maps for space and select a ground map you like best and stick with it. I would recommend Shanty Town (very fun) and Ghost Ship (speed of matches).
PvP organization is a toughy. As another suggestion, due to the current size of our player base, if you are with an organized group suggest to the group leader that you requeue for another match after you complete the one you are in. Most good groups I have seen stay together through many matches and its a way to add to the friends list
Overlapping quests also works well in fleet actions (turret killer, 50 NPCs). Best get a group together before you enter a queque, works well in Zone chat by stating what level you are. Unlike Fed zone chat, you have nearly no spam, and usually get a team of your current level together within a few minutes. From there, run with the wolves, turn in/pick up new missions on the way and don't bother to work a list down. That's what I usually do on Fed side, unlocking new missions after completing lower ones; or get access to other nebulas or whatever; but there's no need for that on KDF side.
Whats the rush? There is nothing waiting for you up there at the top...and no noob or lower ranked players to smush. So you might be better off just enjoying the game and content as journey to experience and not a chore to finish.
First time you do it, you got a wave of 6 squads to kill... when it's done, take the following mission asking you to do 8... you''ll already be at 6/8 so kill 2 more squads... Then... T'ong Cluster your heart out. (with PvP when you feel like it)
at LC6, go to D'kel Cluster... LC7, Azure Nebula. Do 8 Romulan Squads... then 8/10, do 2 more... you'll be LC8...
Return to D'kel...
The Mechanic is to perform every mission possible (PvE and PvP) at least Once before it becomes Repeatable. Every "first time" of a mission awards lots of points.
Since Season 1 Came out, I started over my Klingon and, after 2 evenings of play, i'm LC8 or 9... (my previous character is Cmdr2 after weeks of casual play)
Anyway, don't rush things out... the longer you take to reach cap, the more end-game content will be available to you.