Study your skills
Somewhat obvious and joyously simple best describes this advice: Take a good look at both your Captain's and Bridge Officers' skills. Long term planning will help you both in the early and later areas of the game. Plus, it never hurts to know what you're working towards. In the end, sitting in space for 15 or 30 minutes is worth knowing where your character is headed skill-wise.
I recommend a good balance of space and ground combat skills, although even I have a tendency to favor maxing out space skills first. There's a Starship Operations skill that gives general benefits to various aspects of commanding a starship -- that's a good one to start with. Also, remember that your chosen class will give you a unique set of class-specific ground combat skills. These are worth knowing inside and out, especially if you plan on partaking in any ground PvP battles.
There're skill points in them there Episodes
An Episode mission moves STO's main storyline along in a manner very much like an actual episode of the television series. You'll get a little space combat, a little ground combat and some cool story moments. Some of the best missions are Episode missions, and they give you a nice chunk of skill points while you're distracted by all the cool television-like story moments. You'll find the missions with Admiral Quinn during your stretch as a Lieutenant.
Variety helps, too
Great though they may be, doing nothing but Episode missions could bore some people. I find it helpful to try all the various mission types at least once. Patrol missions are nice when you're looking to do mostly space combat. Sector defense missions require you to complete enemy signal contacts, which will force you to group with other players. This being an MMO, I hope none of you have too much of a problem with that fact.
We've all had our horrible pick-up group experiences, but some of the best times in STO
involve blasting fools with other meatspace players.
The hidden wonders of Transwarp
Need to head back to Earth Starbase? You've got an ability that will get you there in an instant, but for some reason it's hidden within your hot bars. When in your ship, find your hot bar on the default UI setup's right side. There should be a small button on the right end of that row of buttons -- it sort of looks like three pages sitting atop one another. Press that tiny button to bring up three numbers; the third number is the one you want.
Voila! Now you've got three rows of hot bars and you should see an icon hidden within one of these marvelous bars -- that's the Transwarp ability. It takes you back to Earth Starbase, which is always useful when you're ready to buy some new stuff or perhaps get promoted to Lt. Commander.