I have the basic set, plus the River, and we’re looking for something to spice up game night a bit.
Random points:
Really I’d say none. My personal feeling is that the expansions unnecessarily dilute what was a rather clean design. I know people who play Carcassonne with all of the expansions and the games drag on well past the point that they’re welcome.
If you held a gun to my head and demanded that I pick an expansion (besides the handful of “just a few more tiles” ones) I’d go with the original Inns and Cathedrals.
There’s a good list of the expansions and a lot of discussion on the relative merits of each of them at Thoughts on the Carcassonne expansions . It’ll be a lot more help then I’ve been. 
(Geez, if I had found that list on BGG, I probably wouldn’t have posted this.)
For ease of play, the first expansion (Traders and Builders) probably adds the least amount of complication (basically, it adds 1)one “uber-meeple” worth double for control of a city/road/farm and 2)some special road and city tiles that make that road or city worth extra if it completes but nothing if it doesn’t).
As for the issue of games running too long, if you get more than one expansion you might want to randomly take some tiles out so that the total in the game isn’t much more than in the original game.
Correction: That should be “Inns and Cathedrals”. “Traders and Builders” is the second expansion, which adds another wrinkle to farm scoring (which I gather you don’t want) and a way to get two turns in a row (place your “builder” on a city or road where you have a meeple, and get a second turn every time you extend that city or road).
Traders and Builders as well as Inns and Cathedrals will add a lot to the game without bogging it down. Those are about the only two expansions I consider to be must have. I own all the expansions and those two are the only ones I use on a regular basis.
Marc
I agree with MGibson. We have most of the expansions, and Traders & Builders and Inns & Cathedrals are the only ones we use every time we play. The others can add some occasional interest but aren’t really worth buying, IMHO.
Owning most of the expansions, my usual house rules are:
-skip the river
-randomly remove 1/3 of the tiles
-Include the Builder and the Thug (the weird shaped meeple and the oversized meeple)
-Play with the trading goods
-only use the Cathedral/Lake ‘double or nothing rule’ with experienced players
I recently picked up the Princess and Dragon, and the Tower, but have yet to play with them.
I own and have played with all the expansions, including River II and King & Scout and The Tower (Don’t have Die Cathars however :(). If you play with all of them, it makes the game very long, at least with 4 players.
We always play with King & Scout, Inns & Cathedrals, and Traders & Builders. We also use the tiles from The Princess & The Dragon, but we don’t always use the rules for that expansion.
The Count really changes the flavor of the game, making it much more cutthroat than the original. The Dragon also does this, but not quite as much. We don’t care much for The Count, but we do enjoy The Princess & The Dragon. The Tower again changes the flavor, making play much more strategic, especially when and where to place farmers.
We like using the tiles from all the sets, even if we don’t use the rules. We just like having all those tiles to choose from.
My gang has the most fun with the Princess and Dragon set. We act out the Dragon’s chomping sound effects, and keep score of all the people and pigs he’s eaten. 
On a more intellectual note, the dragon makes the game more unpredictable, and frees up farmers who might be trapped in a field the game’s progress has made pointless.
I’m in general agreement with MGibson. Inns and Cathedrals is quite good. Traders and Builders is good. The King is also good, but if you combine King and Traders and Builders, you get to the point where it makes more sense to complete other people’s cities instead of working on your own, which is less fun. The Count, the Princess, and the Tower all suck quite a bit, imho.
The best part of the other sets is simply getting new tiles to play with, and you don’t always have to use the rules associated with them. And yes, randomly pull out 30-40 tiles to shortren the game up. It also adds some spice, as you don’t know for certain how many of a particular shape are left.
Have you tried the Hunters and Gatheres standalone? I consider it a purer form of the game, in fact, we play that a lot more than the original game. Just throw in the Scout expansion (except for the sucktacular Shaman piece) and you’re good to go. 50 minutes, tops.
The biggest thing I can contribute about Carcassonne is the general tile-picking rule we use: Instead of a player, on his turn, flipping over one tile at random and playing it, a player flips over N+1 tiles, where N is the number of players. He plays once of the face-up tiles. The next person plays one, etc., until the original flipper plays the last tile. Then the next person flips over N+1 and gets the first choice, etc.
This adds quite a bit of strategy and defense and control without bogging things down too much.