Use your replicator!
If you look closely at your inventory window, you'll find that there's a button labeled "Replicator" that does something quite awesome. What does it do? Why, it allows you to sell your junk loot anywhere you like! Well, so long as you're in a starship. If you're on an away mission, obviously you can't use the replicator systems present on the big space ship.
Managing your shields
Space combat can get pretty hectic and figuring out the best way to handle your shields is often going to determine the success of particularly challenging fights. There are two basic methods for handling the redistribution of power to your shields. Most people simply click on each individual segment of the shields user interface, which will pull power equally from each of the other three shields. That's well and fine in a simple fight, but when you're being bombarded from various angles -- which is usually the case -- you'll want a better method.
For those times when two or more sides of your shield are taking various levels of beatings, remember that clicking the center of your shield UI (the globe, essentially) will redistribute power equally among the entirety of a ship's shield.
Finding your way around
Navigating space can be a daunting task but thankfully Cryptic's supplied a system list feature. If you press the "M" key, a window with three tabs will open up. One tab holds a top-down map for the sector of space you're in, another has the galaxy map (IE the world map) and finally you'll find a third tab containing an alphabetical list of every destination within your current sector.
Every mission comes with a destination, and assuming you're in the right sector of space (check your galaxy map for that) your system list is going to be a big help. Just find the location, double click it on the list and your ship will auto-navigate to that location. You can also use the visual top-down map to do this as well, but finding your destination via the list is much easier. Of course, you may prefer to go for immersion and avoid the list, although the reality of being a starship captain is that you've got an underling to handle the boring stuff.
I recommend some coffee or tea while you wait for the warp engines to take you where you need to go. Some Earl Grey perhaps?
Tweaking your setup
This is pretty simple, but it's worth mentioning just in case some people disregard it. Both your action bar and your power management window can be adjusted to your preference. I suggest messing with the power window at the very least, as you may find that one of the other variants is more useful to you. The button you'll want to look for resembles three pages stacked upon one another. The three options give you varying levels of UI complexity, depending on how crazy you like your MMOs to get.
Another thing you can tweak is the way your ship looks. Cryptic is known for their customization madness, and things are no different here -- at least for Federation players. Once you've completed the tutorial, you'll find that the Earth Starbase has an NPC who -- upon being spoken to -- will allow the customization of your ship. It's pretty deep and will likely cost you a little bit, although in beta it's been a free service. Still, hopefully by launch the first "refit" will be free.
Be on the lookout for future guides and editorials focused entirely on Star Trek Online