Start out by offering up your LFG services and watch all the level or zone channels for call outs. If other players or roles are calling out, even if you and they are just DPSers, take leadership and start inviting. I can't tell you how many times I just took over on setting up a group for an instance I haven't even been in because others where calling out and no one was starting it. Even if you only had 3 dpsers to start out it shouldn't take too long to find an tank and healer. After you have formed the group you can hand over leadership to someone who has more experience with the instance if you want.
Play with friends or Guildmates. OK, so not everyone is in a guild or has consistent online friends to group up with but trust me, its not that hard to find either in an MMO! I find that if one of my friends or guildmates are online it is much easier to get a group, because we pre-group with more of the necessary roles (tank, healer) but see the point above if you are a loner.
Finally this last bullet is for the people leading/setting up the groups - stop being so darned picky. If you ask for a role and that person volunteers for it, give them the benefit of the doubt. Don't be nitpicky by asking what their spec is, what gear they have, blah, blah, blah. PVE Instance leveling from 1 - 50 doesn't require that much scrutinizing and if you think it does then you probably shouldn't be leading the instance. Let people join, first come first serve and make the best of it. If for some reason someone is really not pulling their weight, understanding your instruction, or playing horribly wrong for their calling then be nice about it and give them some pointers on fixing their game then politely remove them from the party. If they are really sore about it for any reason you can always /ignore them. Later on you can be more serious about the T1/T2 runs, but hopefully by then people are doing it with a trusted group of friend or guildies so then you won't have to worry about pugging.
Thank you for being sane. I have gotten into at least a dozen instances by grabbing th 2 other dpsers going solo and putting a group together looking for only 2 people rather than 3 groups of 4.
I really hate when people believe their group comp way is the only way to complete an instance. Yes sometimes it is the only way to complete an instance in 20 minutes. If you don't spend an hour turning people away you could have done it in an hour and 15 minutes and still have gained time while u looked for your perfect group though.
And on guilds. Sometimes there are issues. If you get on at weird hours you get the awkward all the healers got on 20 minutes ago and got swiped up. But don't be afraid to ask your friends if THEY know somebody. People seem to be afraid to ask their friends to help out.
Don't forget you can add notes to your friends list to remind you what roles they play and how well. (I have a few people on my friends list so that I know to ignore them.)
And on a note. Healers are impossible to get in this game despite the fact that 2 of 4 callings can heal. If you are a cleric or mage and can 'never get into an instance' make a healing role. No, I am not saying you have to heal because you're one of these because you don't but if you refuse to you have no right to complain.