Strong recommendations for
Jaipur - beautiful aesthetics, easy to grasp, fun and addictive (a bit fiddly to set up though). My DC have enjoyed this since they were 8 or so.
Battle Line - this is our favourite. Simple rules and almost unbearable tension.
Port Royal plays very well 2-5, our DC (9 and 12) love this as well.
We used to play Monopoly Deal as both 2-, 3- and 4-player. Brilliant little simple and addictive card game that we must have played hundreds of times.
As for Lost Cities - I love this but DH and the DC find it too dry.
We had Agricola All Creatures Big and Small but sold it. Deeper and more static/strategic than the others recommended above which are more simple and more dynamic/tactical.
We have Patchwork but didn't like the tetris/spatial aspect. Lots of people love it though, so worth considering whether the spatial aspect would appeal or not.
I also love Carcassonne as a tactical 2-player game, but can't get anyone else in the family to play.
This year I have bought Hanamikoji for DH, it is supposedly a brilliant two player game. Possibly a bit deeper than Jaipur, Battle Line and Lost Cities.