Lol, a lot of suggestions and I'll think of adding mine.
RTS
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Sins of a Solar Empire: Amazing game single player. I don't think you can find any sci-fi strategy game that can beat it (Starcraft is somewhat dated nowadays). The only thing that ruins it is multiplayer, where most people insist on ramping the speed up to max and ruining the game.
World In Conflict: Another tough to beat strategy game, this one gets rid of base building. You have a set number of 'requisition points' that you use to buy units. As your units die, you get those points back over time. You pick a specialty: Infantry, Armor, Air (helicopters) and Support (repair, artillery and anti-air). You then play with a team of up to 8 other players to fight the opposing team. You can also call in support using points you get from killing and taking capture points, which you can use to call on: of-map artillery, airstrikes, carpet bombing, para-troopers, napalm, tactical nuke to name a few. I think this is the #1 team game, but you will probably want some friends to play it with. Also, the system requirements may be a bit high.
Medieval II: Total War (or any other Total War game): This is the strategy game that makes you think more than any other. It is almost a war simulation without getting too bogged down with the details. Some decisions, such as where you place your archers, can turn the tide of the battle. I like to position long-bowmen (English special unit) on top of a wall I just captured (during siege) and then use their extended range to fire upon units gathered in the center. If they try to attack me, those same archers will tear them to shreds. I like how one unit can comprise of up to 60 units in formation, making it look real. And some details will make or break you (never charge cavalry into the front of a spear formation, they will instantly die).
Demigod: Made by the same people who made Sins of the Solar Empire, in this, you control one unit called a Demigod and fight alongside your human counterparts with a bunch of smaller AI units. It is kinda like Diablo II mixed up with an RTS game. If you ever played some of the custom games in WarcraftIII, you will know what I am talking about.
FPS:
I only really know good single player FPS.
Bioshock (somewhat high system requirements)
Half Life series
Portal (which is really more of a puzzle game)
Doom3 (can get repetitive tho)
Left 4 Dead (somewhat high system requirements; coop multiplayer)
Prey
And if you want really old.
Star Wars: Dark Forces II
that is like the first FPS I ever played. It's on steam.
