Welp. I'll be honest... Star Trek Online is a very combat oriented game and there are no game mechanics that allow you to side-step combat on a regular basis. You will be fighting and blowing people up in a very un-Starfleet kind of way for the majority of your progression through the story line. The developers have found it difficult to create an MMO that allows level progression without killing things, and that is apparent when you play the game.
HOWEVER, there are elements of the game that allow you to immerse yourself moreso into an RP environment.
They duty officer system: Once you make it to Lieutenant Commander rank, you can send your virtual crew off on assignments throughout the galaxy. They'll do their best to complete those assignments and bring back experience points, commodities and other various goods that will help you with your progression throughout the game. The assignments that are available for your crew change, depending on where in the galaxy you are at the time.
The Admiralty system: Upon becoming a Fleet Admiral, all of the ships that you've previously flown throughout your game progression, become available for you to send out on assignments, much like the duty officer system.
New Romulus: At level 50, you'll gain access to the New Romulus adventure zone which is one of the game's largest zones and is almost entirely non-combat oriented. Possibly the most Starfleet feeling of your exploits will be in this zone, helping researchers, tracking animals, locating artifacts, and learning the secrets of the planet. Some people feel that the lack of combat makes this area boring, but I actually enjoyed it a great deal.
The Foundry: A user generated mission database, some of which are amazing creations that really have more of a Star Trek-esque feel than the official missions that you play to level your character. A listing of some of our best fleet member authored missions is here:
http://fleet.stonewallgaming.net/forums/the-foundry/73967-the-foundry-folder-fleet-mission-indexRoleplay fleets and events: There are some fleets that offer a roleplaying environment for your character. Stonewall's RP is on-again/off-again, but there are other fleets and channels that you can team with for an RP immersive experience.
There are more ways that you can enjoy the game for more of an RP standpoint, but since I'm not much of an RP'er, I'll have to rely on other people to chime in on this thread with additional info.
Hope this answers your questions a bit!
Voleron (Dave)