Oh boy, you're stuck in the World of Warcraft's super lag-fest today. It's 1PM EST and nothing has changed, you can't cast any spells, you can't find any enemies to kill, and you have no clue how you're going to save Azeroth if the forces of lag are holding you back. Well, there are a few ways to deal with the lag and play today as if no one else bought the expansion. We'll go through the easiest to the hardest, to give you as many options as possible to deal with lag in WoW.
You will be experiencing lag today if you're on any of the high-pop or even some of the medium-pop servers.
Find a Friend on a No-Pop Server
If you know anyone that is on a server that has no one on it, then good news for you - have them battle.net invite you. It's going to pull you to their server and you should now have a far less busy instance of the world. You're going to move through the expansion in the fast lane, because the cordination and time to set it up is too much to get the thousands of players into these no-population servers, so you should be able to get through the game without an issue.
Again, the process is easy. Get a friend a server without lag and have them invite you. You may need to /reload or wait until your character full moves over to that instance of the world. Voila. Now, there is some issues to be aware of. Majority in a party usually stays on that server, so two players from Server A in Server B's group will have everyone probably on Server A. Next, if you drop group, you'll return to the land of lag hell. Finally, sometimes it doesn't work for some reason. Try relogging.
Buy A Second WoW Account on a Second Battle.net Account
Got money to spend to avoid lag and want to make Blizzard richer? Do the same thing as above, except buy the starter edition of WoW. Any edition. On a second battle.net account. It should allow you to battle.net friend yourself and invite you over. This could work with the free starter edition, but I'm not 100% sure. You could try that first before you spend any money, but grouping with the free to play players is a bit tough.
Transfer Off a High-Population Server
If you don't have a guild that needs you, you can easily transfer off a high population server. There is no reason to be on a server except economy, which often times the economy of lower pop servers is more favorable than high pop servers and for a guild. Otherwise, everything you could do you could do through battle.net, including raiding. Heck, you don't even have to be in a guild to raid with a guild. A lot of guilds are totally fine recruiting off-server and just having the raid leader battle.net invite people.
That's a more permanent solution to the issue, but the good news is that you won't have to lag in Stormwind / Orgrimmar anymore.
Wait it Out
You can always wait until next week, which the huge rush will pass. Otherwise, these are your only options for the short-term.
Annex: Broken Quests
I was stuck on a quest like Gormaul Tower, I just simply logged out and exited the phase and walked back in, and it seemed to work. Many broken phases can be fixed by making sure you're on YOUR server, then logging out / in, and then reentering the phase.