Hi NazcaC2!
I enjoy various team cooperative games where some sort of strategy is required and no man is left behind.
Some examples are Borderlands (Xbox360), Army of Two (Xbox360), Trine (PC), Left 4 Dead 2 (PC), The First Templar (PC), Team Fortress 2 (PC), Vindictus (PC), Arma2 - MODs (PC), Fallen Earth (PC), Might & Magic: Heroes - various (PC).
However, I am not a big fan of fragging - PVP, because it becomes repetitive and lacks the aspect of story telling, exploration, and adventure.
When it comes to solo gaming, I seek games with an intriguing story and sense of exploration and adventure.
Some examples are Alien Breed - Trilogy (PC), The First Templar (PC), FreeCiv (PC, Android), Omikron (PC), Blade Runner (PC), Fallen Earth (PC), Might & Magic: Heroes - various (PC), Metro 2033 (PC), Pokemon Soul Silver (DS), The Legend of Zelda: A Link Between Worlds (3DS), Dragon Age (Xbox360), Half-Life (PC), Dark Messiah of Might and Magic (PC), FTL: Faster Than Light (PC, Web), The Dream Machine (PC).
From those selections, you can tell that I am inclined towards Medieval and Sci-fi genres, and post-apocalypses (if done right). :D