Personally, my favorite expansion is Prosperity. My least favorite expansion is Alchemy.
So, if I had to suggest an order, it would be:
Prosperity, Seaside, Intrigue, Alchemy.
Here are the major new things in each set:
Multi-type cards (ex. Action – 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
Multi-turn actions (called Durations)
New types of Attack cards
Tokens and Mats for use with certain cards
Emphasis on setting up your next turn
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
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