Kill the Foozle: There’s a foozle over in that tower/dungeon/penthouse, and you must kill it. That specific foozle, not just any old foozle.
Slaughter the Rats: There’s a bunch of rats (or other relatively non-threatening critters or people) somewhere around here. They need killin’, whether it’s to thin the numbers, send a message to the survivors, or because someone needs their duodenums for some daft reason. Mysteriously, in the latter case, only a small percentage of them actually have duodenums. In some cases, only the rats in one very specific location will do.
Follow the Breadcrumbs: Something–a trail, a mysterious letter, a bombshell who gives you a Significant Look before slipping away, whatever–leads you to a new area.
Touch Base: Meet a contact. If it’s a new contact, you may Follow the Breadcrumbs to them.
Rescue the Princess: Find and free someone (actual royal blood optional). The good news is that once they’re loose, they can make their own way home.
ESCORT the Princess: A whole different kettle of very smelly fish. Follow the very, very stupid, danger-prone princess as she wanders slowly and aimlessly through dangerous areas, getting beat up on her behalf, until she becomes someone else’s problem. (More on this later.)
Fetch the MacGuffin: Go get the thing. You know, the superweapon/secret documents/magic ring/boss’s lunch.
Deliver the MacGuffin: FedEx quest. Now that you’ve fetched the thing from Place X, take it to Place Y.
King of the Hill: This location (despite its utter lack of hardpoints, choke points, or any other point to suggest tactical value) must be defended against waves of attackers. (Good point defense missions actually give you those points, and/or the means to construct them.) Also, sometimes you are the Hill, and you’re just trying to survive the waves where they’re coming after you.
Face the Champ: Engage in a contest with the local champ and beat them at their own game. Single combat, a race, a drinking contest…
Explore (the Fog of War): Go to a place and look around. Frequently turns out to be a different quest type in disguise, but sometimes it really plays out as reconnaissance.
Infiltrate the Base: Sneak in someplace and do sneaky things sneakily. Frequently involves a Foozle or MacGuffin, but sometimes it can be free-roaming sabotage or general burglary.
Now, as to escort missions…hoo boy.
In fairness, I will say that they can be done well, for values of “well” approaching “less annoyingly”. Some City of Heroes escort missions weren’t bad, since the escortees mostly followed the player, rather than going on their own predetermined path. You generally couldn’t leave them in a hiding spot and go clear a path, but you could hide with them on a side path while a patrol passed. Giving some of the escortees powers and perception stats provided more personality, too. (I found myself liking Fusionette for her zippy, gung-ho, suicidal behavior, because it was a refreshing change from the plodding, dumb, suicidal behavior of other escortees.)
The single most annoying escort mission I recall was in LotRO. Lalia Holeinherhead (or somesuch) was lost in the Barrow-downs, and you had to go save her. The fact that she is ostensibly terrified does not stop her from moving at a very slow walk, insisting on searching for her cloak, or plodding straight into ambush after ambush until you have walked every inch of this place of death. The only thing dumber than Lalia is the player character, who knows the way out and could just pick her up and walk straight out with her, but inexplicably indulges her. Also, she was out there looking to pick up a guy, which is saved from going beyond Goth to outright necrophilia only because she’s too dim to realize that her celebrity crush is long dead.