How to teach strategic thinking to non-gamer players?
Talk about it.

That's it; make the game a topic of conversation. Talk about it at random moments. Talk about your strategy when you're breaking out the game to play again. Talk about other players' moves while they're moving, and what plans they might be laying. Talk about how to break up an opponent's strategy, or even how to break up your own strategy. When the game ends, talk about how the winner pulled it off.

You should try to avoid boring the other people to death, of course, but the mere act of hearing about how you can plan for (some particular game) will put more tools in their toolbox the next time they play. These "non-strategic" players haven't put much thought into how to win these games, and probably don't know how you can do long-term planning for those games. Most games have a particular strategy that works well; some work best with purely-tactical moves, some work best when you've planned your strategy 15 turns ahead, some work with a mix of strategic planning and proper tactics as opportunities crop up. If you really want these players to improve, and they haven't figured it out themselves, share your hard-earned insights into the game. Don't keep your secret game-winning strategy to yourself just because you're a knowledge miser.

To illustrate, here are a few examples of things I didn't even realize were important until long after I started playing that particular game. (I make no claims to being a genius.) Withholding key insights from games can really cripple someone's ability to play well. Share some insights like these, and they'll start seeing how the game is more subtle than it initially looks.

Puerto Rico: you can screw other players by choosing Captain to force players to ship their valuable trade goods, and Craftsman is frequently a bad move.

Glory to Rome: it's a bad idea to play Craftsman cards to lead the Craftsman role, because the player to your left will Patron, and pick up a Craftsman client. Playing Craftsman cards to follow the role is only a good move if the player to your right led the Craftsman action.

Dominion: Coppers are bad, and should be actively removed if reasonably possible. If you want to afford Provinces, you really have to get your average-coins-per-card up.

Race for the Galaxy: I still don't understand how to do a proper mix of military and trading strategies. Sure, there are plenty of developments that support both, and no military tableau should purely be military developments and military worlds, but the major source of victory points for a military strategy is the high-value military worlds, right?

Power Grid: Don't buy a power plant every round; only do it when it's a significant improvement over what you already have.