In my experience, being in the fleet and going "/1
[my class] LFG [name of flashpoint]" works reasonably well.
Alternately, paying attention to General and responding when someone goes "[flashpoint] group LF[number]M" for much the same reason.
EDIT :
Update 1.3 introduced the Group Finder Queue , and gives out bonus commendations once a day for using it.
Learn it.
Use it.
The more people who use it well, the better it works.
gaming.stackexchange.com/users/7864/Shadur
Gather in the flashpoint location and flag yourself in /who or in the Social window (Press "O") as LFG, you can also leave a short note there (E.g., "LFG Hammer" or just "hammer").
You can then search for people with LFG tags.
It's nice tool, but it's a shame not many people use it.
gaming.stackexchange.com/users/15772/Krazer
Most people post in general, so I would just do it.
It's effective.
Sometimes you'll also find people waiting outside the start area for the quest, and you can invite them or vice-versa.
It's particularly unambiguous if you or they have the purple LFG icon hovering above you by setting your status in the /who list, as Krazer notes.
I've noticed that most people don't use the LFG flag, unfortunately.. See Andrzej's answer for a description of the new Group Finder tool, which is another option.
gaming.stackexchange.com/users/6066/Matthew Read
With the recent addition of patch 1.3, the Group Finder is by far the easiest/most effective way to get groups together for flashpoints (both heroic and normal), story-mode Operations, and heroic quests on planets.
It's available from level 10 onwards as a tab next to the minimap.
See the linked official guide for full details.
gaming.stackexchange.com/users/3720/Andrzej Doyle
Posting in channels can be an effective way of finding groups (though not as ideal as an automatic queueing system or a commonly used /who system).
Other games that do have these more effective systems still manage groups with this method.
That said, a piece of advice: If you are partied with no one else, but are willing to start a party, post with "LFM" instead of "LFG".
People will be more likely to send you a tell if they too are looking.
gaming.stackexchange.com/users/18334/Mezner
So I just got my first empire Flashpoint, a 2 player job. Since it starts from the fleet, I don't really want to just ask in fleet general chat if anyone wants to do the quest with me. Is there a better way to find groups for Flashpoints? Such as a built in system that I am missing?
What is the most common method to find a group for a Flashpoint?
gaming.stackexchange.com
<star-wars-the-old-republic>
gaming.stackexchange.com/questions/45163
gaming.stackexchange.com/users/4393/Resorath
Try the Group Finder Queue which is the easiest way to get groups together.  While most in the community tend to agree, regarding the ways of finding groups, it seems that the solution someone picks, is hit or miss.
It is mentioned that the client supports an in game way to find groups. Some people tend to believe that using the social mechanics of the game, might help finding a group.