Sit around in Dawn 
Okay, more seriously basically it comes down to a few things:
Finding places people look for fighters, around the Dawnish Glory square is one, around the hospital means if they are short you can slip in easily. (I have no clue for other nations)
If there isn’t one in your nation, try to make one, anything which makes it easier to grab bodies in a rush from when I organise quests is a bonus. If your group tent is tucked away at the back of the Navarr woods, it’s a bad place to wait.
Also if you are a small and unknown group, make connections to a better known group, quite possibly spend some time hanging out with them.
Speed in getting down to the gate
It’s pretty difficult to get exactly the right number, even if you have all the time in the world people can find other commitments, loose track of time etc.
Being down there early and keen is going to help with that.
If you are in a group have one person press for more information while the others get armoured up.
Versatility
Fighters are good, Warrior Physiks are better, Combat exorcists etc, there was a period I could get on all the Dawnish quests by having that combination of Physik and wearing armour (It’s less the case now, but might be for other nations.)
You have another skill that might be useful though the gate (Physik and the Priest ones are obvious.)
Make Friends
If you keep seeing the same three faces organising skirmishes, they are a good three people to get to know, be helpful to them, and let them know that it’s a bit of game you are interested in. Also offer to help out, aside from the oppertunity that hafd 4 different groups orgainising it, I’ve never failed to get on a quest I have helped orgainise.
Finally try not to get disheartened if you fail to get on one.
Don’t Drink
If getting on quests is your number one priority the least oversubscribed ones are late Saturday night, particularly the events closer to the OOC Summer Solstice .
(Finally finishing the post I started two months ago, but it doesn’t seem a bad time to raise the thread)