I have the Dominion base set only, in what order should I buy the expansions?
Personally, my favorite expansion is Prosperity.  My least favorite expansion is Alchemy.

So, if I had to suggest an order, it would be:
Prosperity, Dark Ages, Seaside, Intrigue, Hinterlands, Cornucopia, Guilds, Alchemy.

Here are the major new things in each set:

Intrigue

New Mechanic: Multi-type cards (ex. Action &ndash; Victory)
Actions tends to be more complicated than Base's actions
Nastier attack cards
Standalone (doesn't require Base)
Emphasis on attacks and special actions

Seaside

New Mechanic: Durations (Multi-turn actions)
New types of Attack cards
Tokens and Mats for use with certain cards
Emphasis on setting up your next turn

Alchemy

Smaller set (12 Kingdom cards instead of 25)
New basic Treasure card: Potion, costs $4 worth 1p
Powerful cards, but 10 of 12 take a Potion to buy
Emphasis on extra actions/action chaining  

Prosperity

New basic Treasure Card: Platinum, costs $9 worth $5
New basic Victory Card: Colony, costs $11 worth 10vp
8 Kingdom Treasure cards that act like actions when played during Buy phase
New Victory tokens, worth 1vp or 5vp each
Emphasis on treasure and victory points

Cornucopia

Smaller set (12 Kingdom cards instead of 25)
Emphasis on Actions and Drawing cards
Doesn't mix well with Prosperity
Jester is the real star of this set as it lets you gain one of a card an opponent flips up.

Hinterlands

Several new Kingdom Victory cards with special effects
New Mechanic: On gain, do X
Emphasis on +Cards and gaining cards
Puts a new twist on some existing card ideas (ex. Margrave instead of Militia)

Dark Ages

More new Kingdom cards than any other set
Shelters provide an alternative to starting with 3 Estates
New card type: Ruins. An action card that isn't very useful (brown border)
Emphasis on giving other players junk and action chaining

Guilds

Smaller set (12 Kingdom cards instead of 25)
New Mechanic: Coin tokens that you gain and can spend later
Emphasis on trashing cards and gaining coin tokens