The key to surviving a PuG with your sanity intact is your mindset going in. Don't expect an efficient, well-oiled machine. Things are going to take a while to start up, and it's going to take a few minutes for players to get used to one another's play styles. As weird as it sounds, the best raid/party leaders aren't necessarily the ones who are nice to everybody. They're the ones who push just a bit harder than you want.
PuG-ing raids and dungeons are two, completely different animals. Particularly if you're pugging a raid, things are going to take a while. This should be expected. However, there is definitely a fine line between normal downtime and having to wait half an hour between attempts on Patchwerk. In fact, if your raid is wiping on old Patchy, there's probably something wrong. A key skill is reading the signs on how the raid is progressing. As long as it's constant, things will probably be okay. If things are stop and go, stop and go, you might want to consider (this is important) POLITELY bowing out.
Dungeons don't take nearly as long as raids to put together, but they do seem to fall apart much easier. A heroic group wipes once and all the sudden the healer's like 'DONE!' and leaves the group and hearths out. Then all the rest of us look at each other and shrug sadly, and I make a portal to Dalaran. People just don't mind leaving a dungeon group as much, perhaps because they know it's not all that hard to find one. People don't leave raids because they've been saved to that raid; they can't just go join another in trade chat. It's hard to say how to aid this problem. If you get in a group and the tank or healer is inexperienced, the group is going to fall apart really fast, no matter what you do as a bystander. The only thing in that situation is to be positive, and be confident with the rest of the group that you can find someone else to replace the noob. Otherwise, you're pretty much stuck.
The other possible piece of advice is obvious. Take some friends into heroics. Even if you wipe a few times, friends don't bail on friends. Or, go with some guildies. Especially in a larger guild, there's sure to be at least one who'll help you out. And above all, if you do find a good group, whether in LFG or with friends, run more than one instance with them. You can do two or three instances with a good group in the time it would've taken you to do one with a bad group. Also, be polite while you're in the group. That one jackass in the group just makes everyone want to quit and log off for a while to find something to take their aggression out on. Which isn't good for progression. Be polite, and be effective at your job, and someone will invite you back for another round of boss-beating fun.
Overall, just be patient with this whole ridiculous business. Just think about the goal: get enough decent blues to not be a total hinderance in a raid. Find a good guild, get a raid schedule going, and pretty soon, you'll be in some shiny new purples, ready for Ulduar.