We really do need to sticky one of these…
This is the most popular question on these boards, and you'll get different answers from everyone.
Here's my stock answer:
If you can get a big-box right out of the gate, GET DUNWICH HORROR. The board's lousy, the Horror itself doesn't activate that often but it's got some great Investigators, Monsters, and Ancient Ones - And Madness and Injuries. After you play with Madness and Injuries once, you will never want to play without them again. Period. Of all the Expansions, big and small box alike, Madness and Injuries are by far the best single addition to the game. You will use them every game, and the thought of abandoning them will never once enter your head. They are THAT CRITICAL.
If you want a Small Box, you have four to choose from, and all have their strengths and weaknesses.
The King in Yellow: This one is generally held to be the favorite, and it's my personal favorite as well. It doesn't add a lot to every game (if you don't use the Herald or Acts in every game) but it is by far the most Thematic and Intense, and it's Herald is easily one of the two best. If you want a distinct flavor to your game, this one is the one to get.
Curse of the Dark Pharaoh: This expansion is solidly thematic, and the recent revised edition makes its Exhibit mechanic interesting. However, it also includes the single worst Herald they put out, and the Patrolled mechanic isn't that fun. If you ignore the Herald (I do - I've never played with it and I never will), the Whispers mechanic and the solid theme make it a worthwhile experience.
Lurker at the THreshold: Lurker would be my number-two choice behind King in Yellow. There are many complaints about the exploitability of the Herald, but it's also the most interesting Herald (not the hardest or most fun, but most interesting). Relationships add a little flavor, but we rarely use them. But another good bump are the Lurker gates (which have abilities) and it also features some Gate Bursts to make your games a bit more threatening. However, it's the least thematic expansion - just mix the whole thing to every game and you won't notice a theme, but it'll improve the experience.
Black Goat of the Woods: Get this expansion if you hate your gaming group. The Herald is the hardest (YIPES) and the mythos deck is darn mean too. It comes with a lot of monsters, most of which are Dark Young which just suck to fight. But it's mechanic is useless - it's incredibly difficult to find the Cult, much less join it, and the other cards will get virtually no use. This is widely held to be the weakest of all expansions, and I'd advise you to get it last of the small boxes.
As for the other big boxes:?
Innsmouth - Innsmouth has some fantastic Investigators, and Personal Stories, which are not as critical as Madness and Injuries, but we use them every game as well. It's got some terrifying Ancient Ones (Quachill Uttaus is by far my favorite, even though we've never beaten the dusty bastard), and Innsmouth itself is fairly interesting and very hostile. I'd recommend it as the second Big-Box pickup after Dunwich - you'll get a few Epic Battle cards you can't use until you get Kingsport, but you won't miss anything.
Kingsport: Except for (maybe) Miskatonic or Black Goat, I'd get this expansion last. The Epic Battles are cool in theory, but you get to that point very rarely, and the Sinister Plot cards are practically a waste of space. The Kingsport board has an interesting mechanic, but so little goes on in Kingsport itself that whoever is stuck with Kingsport duty will be hating life by the end of it. Two of the three guardians can be fun, but they aren't necessary. And honestly, the best thing to come out of that expansion is the Tulszcha Herald, who is up there with the Lurker and the King in Yellow for most interesting and fun Herald. The Ancient Ones and Investigators are ok as well. But still, of all the Big-Box sets, this one I felt was the least necessary.
Miskatonic: Don't get Miskatonic until you have most the expansions. You don't necessarily have to get it last, but you should at least have 2 of the Big Boxes and 3 of the Small Boxes before it becomes worth it. But when you do, it is very worth it. Every little annoyance or issue with dilution, ect. is well handled in the expansion, and it will bring new life to your expansions.
So there's the run down. If you want to know the order to buy them, I'd recommend:
King in Yellow and/or Curse of Dark Pharaoh and/or Lurker at the Threshold
Finish off King/Pharaoh/Lurker
Black Goat of the Woods
Hope that helps!