Oooh, I have many opinions on this subject. A few recommendations in no particular order:
Dominion. Each game uses a different combination of 10 card sets drawn from many sets in the base game (and there are many more added card types in expansions) so it has incredible variations in strategy possible from one game to another depending on which card sets are included. I realize that sounds confusing, but once you play it you get the basic concept very fast. A lot of people play it as 1 on 1 dueling game but to my mind it’s better with more players.
Pandemic if you like cooperative games - you’re fighting to cure/eradicate diseases around the world, and each player character has unique abilities; however it’s limited to 4 players.
Seven Wonders - excellent game, can be played with anywhere from 3 to 7 players; it takes a fixed number of rounds per game, so it’s good to have as one that won’t drag on forever. A lot of fun, many different strategies which can work in one game or another.
Lots of games from James Ernest’s Cheapass Games - we have the old cheapo monochrome cheap card stock versions of a number of them, and they deliver more fun than a lot of big-budget fancy games. Some of them have been reissued in nicer sets more recently. ‘Give Me the Brain’ is incredibly fun, everyone always has a riot playing it. ‘Kill Dr. Lucky’ is always fun, ‘Deadwood’ is also fun but takes longer, and I know I’m missing a couple other former favorites.
I also recommend Ticket to Ride and Carcassone as mentioned above.
All of the above are aimed at adults but eminently playable by bright kids; if they can handle Catan and Munchkin, they can handle these.
