Many subjects have fundamental ideas that everything builds upon. Boardgaming has many pillars, but I submit that there are 5 major pillars: Alhambra, Carcassonne, Dominion, Settlers of Catan, and Ticket to Ride.
I'm aware that this list is missing a war-game, a dice game, and a cooperative game. If you want those, I'd suggest Memoir '44, King of Tokyo, and Pandemic respectively.
Each of these games brings a little something different to the table. They introduce core concepts that other games build upon, but they are also good games in their own right.
Alhambra - A bit of set collection, understanding where you can score points, even if it isn't first place, and a reward for spacial thinking and planning ahead.
Carcassonne - Very light worker placement, multiple ways to score points, short vs long term investments, as well as learning when to work with other players on a feature and when to cut your losses.
Dominion - Deck building, learning how to maximize potential and minimize "dead" hands, teaches players when to go for more stuff, and when to switch to victory points.
Catan - How to manage resources, trading with other players and the board.
Ticket to Ride - Emphasis on route planning, adapting to unexpected changes (blocking), and balancing what you have versus what you might get.
So there you go, the 5 pillars of boardgames. Agree or disagree? Leave a comment.