We could organize that sir.
As far as board game recommendations... I'm grouping these into three kind of loose categories. Basically Simple Play Strategy are ones that I think are super easy to learn and play quickly, but still require a fair bit of strategy in order to win. Straightforward Strategy Heavy are games that I don't think are particularly unconventional, and may have more parts or take more time to learn or play, but they rely heavily on strategy in order to win. Unconventional, but still strategic games are ones that have a very different kind of game play or some aspect that I don't feel like fit it well into the other categories, but they are still are pretty strong strategy games.
Simple Play Strategy:
7 Wonders (2-7, best with 3 or more) - Very simple, lots of strategy. Lots of fun for large groups.
Dominion (2-4) - Deck building, quick play but loads of fun. They have quite a few expansions too. This is one of my all time favorites.
Splendor (2-4) - Very quick game play. Very easy to learn. But very competitive, be ready to screw each other over in this one.
Forbidden Desert or Forbidden Island - These are cooperative. Both very simple play and quick play time, but there are deceptively difficult.
Straightforward Strategy Heavy:
Power Grid (2-6) - strategy heavy
Castles of Burgundy (2-4) - Lot of pieces and setup, but it's not too hard to pick up on and is one of our favorites.
Pandemic (2-4) - Cooperative play, and if you like the standard Pandemic then try Pandemic Legacy which is fantastic as well.
Ticket to Ride (2-5) - Easy to play, not too long but not quick play either. This is a pretty standard game in most board gamers library and there are a lot of variants too.
Unconventional, but still strategic:
Firefly (1-4?) - This game is absolutely fantastic. Most spin offs from TV shows or movies wouldn't get more than a Meh from me, but Firefly is just so perfect. However, it has a LOT of pieces and it takes up a LOT of space. It also takes a long time to play, 2 hours at least. There are also like four to six expansions that you can add, this will expand the game to up to 6 players I believe and add a lot more variation options onto the game. But I cannot give this game enough of a rave review.
Time Stories (2-4) - This is a cooperative game, but it's unconventional for a couple reasons. It set out to encapsulate the feel of roleplaying, and does it very very well. The second reason is that it is a single story you play through. The base game contains the board and pieces needed and one story. The expansions provide new stories to play. Once you've played through the story it doesn't hold the same replay value for you. That said, it's also VERY difficult. We have played our base game four times and still haven't beat it yet, so... take that as you will. We really enjoy it though. Play time can get quite long too, so make sure you have a chunk of time when you start. And be open to playing it up with crazy voices and put some fun into the roleplay aspect of it. It makes it awesome.
Alchemists (2-5) - This is like Clue crossed with a resource management game. And it uses an app, so every game is randomized and it's really interesting. We still are learning to find the balance in our strategy in order to get a win. It's easy in this one to get obsessed with figuring out the elements, and not get enough reputation points. So it's hard to find the balance in order to win in six rounds. It's got a lot going on in it. Can be a little more overwhelming to learn too, but worth it once you do. Please note that if you play with 5, one of you has to play like game director and you wouldn't use the app. Only four people can compete against each other.